#300119

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	#62529	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1094 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ehi,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, 207315);