#152647

order	512
nilpotency class	5
isoclinism	isoclinic class of #152647
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13059
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33611	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=di,e^b=ei,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224990, #72224994, #72225004, …
maximal normal subgroup of	#72224990, #72224994, #72225004, …
maximal quotient of	C₂×#152647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59843);