#6466084

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10,14,19)(11,20,24,16)(12,17,26,31)(13,15,22,18)(21,30,35,27)(23,33,37,29)(25,28,36,32)(34,39,40,38);(10,15)(11,21)(12,25)(13,22)(16,20)(17,31)(18,19)(23,37)(24,35)(26,36)(29,39)(33,38);(1,3,4,8)(2,5,6,7)(9,11,22,35,14,24,13,21)(10,16,18,30,19,20,15,27)(12,23,36,40,26,37,25,34)(17,29,32,39,31,33,28,38);(9,12,22,36,14,26,13,25)(10,17,18,32,19,31,15,28)(11,23,35,40,24,37,21,34)(16,29,30,39,20,33,27,38);(11,24)(16,20)(21,35)(23,37)(27,30)(29,33)(34,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320528, #72320632, #72320791
maximal normal subgroup of	#72320528, #72320632, #72320791
maximal quotient of	#72322967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373280);