#6874178

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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6781374);