#331710

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,22,32)(20,30,21,31)(25,29,26,28);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(14,16)(17,19,23,22)(18,24,27,32)(20,28,21,29)(25,31,26,30);(9,10,12,14)(11,16,15,13)(17,20,23,21)(18,25,27,26)(19,28,22,29)(24,31,32,30);(2,6)(3,8)(17,21,23,20)(18,26,27,25)(19,29,22,28)(24,30,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238906);