#331830

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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)(10,13)(14,16)(17,18)(19,27)(20,30)(21,31)(22,26)(23,24)(25,28)(29,32);(1,3)(2,5)(4,8)(6,7)(10,14)(12,15)(13,16)(17,19,22,23)(18,24,26,27)(20,28,21,29)(25,30,32,31);(9,10,11,13)(12,16,15,14)(17,20,22,21)(18,25,26,32)(19,28,23,29)(24,30,27,31);(2,6)(3,8)(17,21,22,20)(18,25,26,32)(19,29,23,28)(24,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239026);