#6770905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi

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, 6678101);