#499175

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1170 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)(3,4)(9,10)(11,12)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24);(5,6)(7,8)(9,11)(10,17)(12,20)(13,16)(14,15)(18,21)(22,23);(2,4)(6,8)(9,12,13,18,23,24,22,21,16,20,11,10,15,19,14,17);(10,18)(12,20)(17,21)(19,24)

Polycyclic presentation:

2^9|c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406371);