#312043

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	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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)(10,13)(14,16)(17,18,20,24,21,25,29,22)(19,26,27,31,28,32,30,23);(2,5)(6,7)(10,14)(12,15)(13,16)(18,22)(20,29)(23,26)(24,25)(27,30)(31,32);(9,10)(11,13)(12,16)(14,15)(17,19,21,28)(18,23,25,31)(20,30,29,27)(22,26,24,32);(1,3,4,8)(2,6,5,7)(18,22)(20,29)(23,26)(24,25)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=efhi,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, 219239);