#312138

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 219334);