#363933

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,4)(5,6,7,8)(17,18)(19,26)(20,23)(21,25)(22,27)(24,30)(28,32)(29,31);(9,10,12,13)(11,14,15,16)(17,19,21,28)(18,22,25,31)(20,29,30,27)(23,26,24,32);(2,4)(10,13)(14,16);(9,11,12,15)(10,14,13,16)(18,23)(20,30)(22,26)(24,25)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264809, #72264812, #72264835, …
maximal normal subgroup of	#72264809, #72264812, #72264835, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271129);