#300235

order	512
nilpotency class	3
isoclinism	isoclinic class of #300205
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

1176 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,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,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207431);