#153598

order	512
nilpotency class	6
isoclinism	isoclinic class of #153597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13654
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	#13742	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397 ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

117 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 117 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(2,9,11,34)(3,15)(4,19)(5,24)(6,26)(7,16)(10,38,29,36)(12,43,41,57)(13,32,42,39)(14,45,30,37)(17,50)(18,28)(20,25,35,33)(21,59)(22,60)(23,31)(27,47)(40,61,64,58)(44,56,63,51)(46,52)(48,55)(49,62)(53,54);(1,2)(3,10)(4,20)(5,25)(6,29)(7,30)(8,11)(9,31)(12,24)(13,26)(14,16)(15,42)(17,51)(18,41)(19,34)(21,56)(22,61)(23,35)(27,40)(28,33)(32,52)(36,50)(37,55)(38,59)(39,53)(43,46)(44,47)(45,60)(48,58)(49,64)(54,57)(62,63);(1,3,16,26,8,6,7,15)(2,10,14,13,11,29,30,42)(4,21,48,46,23,17,22,54)(5,18,49,62,28,24,27,47)(9,32,37,38,34,39,45,36)(12,40,44,25,41,64,63,33)(19,53,60,50,31,52,55,59)(20,56,58,43,35,51,61,57)

Polycyclic presentation:

2^9|a^2=e,c^2=ehi,d^2=fgi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,f^b=fg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224652, #72225424, #72225500, …
maximal normal subgroup of	#72224652, #72225424, #72225500, …
maximal quotient of	#72230701, C₂×#153598, C₃×#153598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60794);