#312191

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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,5)(3,7,8,6)(9,10,16,12,15,13,14,11);(2,5)(6,7)(9,11,15,12)(10,16,13,14)(18,22)(19,20)(23,24);(9,12,15,11)(10,14,13,16)(17,18)(19,22)(20,24)(21,23);(1,3,4,8)(2,6,5,7)(9,13)(10,15)(11,14)(12,16)(17,19,21,20)(18,22,23,24)

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=h,d^2=efhi,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, 219387);