#6769719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6676915);