#311131

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18)(19,26)(20,25)(21,31)(22,28)(23,27)(24,29)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(13,16)(17,19,22,20,24,30,23,21)(18,25,27,26,29,31,28,32);(1,4,5,3)(2,6,7,8)(9,10)(11,13)(12,16)(14,15)(17,20,22,30,24,21,23,19)(18,26,27,31,29,32,28,25);(2,7)(6,8)(17,21,23,30,24,20,22,19)(18,26,28,25,29,32,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 218327);