#6808774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3061	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,24,28)(19,25,31,42)(20,23,27,22)(21,34,38,26)(29,45,46,40)(30,32,39,41)(33,36,44,37)(35,47,48,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,32)(21,35)(22,30)(23,39)(24,31)(26,43)(27,41)(28,42)(29,37)(33,45)(34,47)(36,46)(38,48)(40,44);(17,20,23,18,24,27,22,28)(19,25,30,32,31,42,39,41)(21,33,37,26,38,44,36,34)(29,40,35,43,46,45,48,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(3,11)(6,9)(10,16)(14,15)(17,21,22,36,24,38,23,37)(18,26,20,33,28,34,27,44)(19,29,39,48,31,46,30,35)(25,40,41,47,42,45,32,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,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, 6715970);