#300098

order	512
nilpotency class	3
isoclinism	isoclinic class of #300077
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
prime radical	#18640	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207294);