#331667

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	#24238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 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)(9,10,12,15)(11,13,16,14)(17,18)(19,24)(20,28)(21,23)(22,25)(26,32)(27,30)(29,31);(1,3)(2,5)(4,8)(6,7)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(10,13)(11,16)(14,15)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(2,6)(3,8)(10,13)(11,16)(14,15)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32)

Polycyclic presentation:

2^9|a^2=h,d^2=i,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, 238863);