#6749530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,33)(21,27)(22,26)(23,37)(24,30)(25,35)(29,45)(31,44)(32,40)(34,48)(36,43)(38,42)(39,47)(41,46);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,26,28)(20,34,36,29)(21,31,23,32)(25,41,42,39)(27,40,37,44)(33,45,43,48)(35,47,38,46);(11,15)(13,16)(17,20,22,36)(18,25,26,42)(19,29,30,34)(21,35,23,38)(24,39,28,41)(27,43,37,33)(31,46,32,47)(40,45,44,48);(1,4,6,5)(2,7,3,8)(18,26)(19,30)(25,42)(27,37)(29,34)(31,32)(33,43)(46,47);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,21,22,23)(18,27,26,37)(19,31,30,32)(20,35,36,38)(24,40,28,44)(25,43,42,33)(29,46,34,47)(39,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6656726);