#300452

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207648);