#306325

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

677 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)(3,6)(4,8)(5,7)(17,18)(19,26)(20,24)(21,23)(22,27)(25,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(14,16)(17,19,21,28)(18,22,23,31)(20,29,30,27)(24,26,25,32);(1,4,5,3)(2,6,7,8)(9,10,12,14)(11,16,15,13)(18,23)(22,31)(24,25)(26,32);(18,24)(20,30)(22,26)(23,25)(27,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 213521);