#312178

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
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	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=efi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219374);