#6464094

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

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, 6371290);