#151024

order	512
nilpotency class	5
isoclinism	isoclinic class of #7457
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1368
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	#346	C₂³
socle	C₂	C₂×#1368
prime radical	C₂³	#365
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ #346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

137 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 137 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,14,17,37,19,39,15,36)(4,10)(5,22,23,50)(6,26)(7,29)(8,13)(9,16,30,40,32,43,56,44)(11,24,34,51)(12,27)(18,46,45,38,47,61,41,62)(20,48,25,53)(21,35)(28,55)(31,58,54,42,59,63,60,64)(33,49,57,52);(1,3,5,16,6,18,24,42,7,17,25,44,27,45,52,58,8,19,23,43,28,47,51,64,4,15,20,40,21,41,49,63)(2,9,11,14,12,31,22,38,10,30,33,36,26,54,48,46,13,32,34,39,35,59,50,62,29,56,57,37,55,60,53,61);(1,4,8,7)(2,10,13,29)(3,15,19,17)(5,20,23,25)(6,21,28,27)(9,30,32,56)(11,33,34,57)(12,26,35,55)(14,36,39,37)(16,40,43,44)(18,41,47,45)(22,48,50,53)(24,49,51,52)(31,54,59,60)(38,46,62,61)(42,63,64,58)

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=i,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=ehi,g^b=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223646, #72223659, #72223681, …
maximal normal subgroup of	#72223646, #72223659, #72223681, …
maximal quotient of	C₄×#7457, C₂×#151024, #72231924, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58220);