#331828

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	#28231	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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)(10,14)(11,12)(15,16)(17,18)(19,26)(20,29)(21,23)(22,25)(24,28)(27,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,12,15,16)(18,23)(19,27)(20,22)(24,31)(26,32)(28,30);(9,11)(10,12)(13,15)(14,16)(17,19,21,28)(18,24,23,26)(20,27,22,30)(25,31,29,32);(2,6)(3,8)(9,12,13,16)(10,15,14,11)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,31,26,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=hi,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, 239024);