#433482

order	512
nilpotency class	3
isoclinism	isoclinic class of #419463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15481
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237063, #72237078, #72237094, …
maximal normal subgroup of	#72237063, #72237078, #72237094, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340678);