#377500

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ef,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284696);