#6718457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#29725	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 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)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(18,23)(19,27)(25,37)(26,34)(29,42)(30,35)(40,41)(44,46);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,18)(19,28)(20,25)(21,23)(22,34)(24,30)(26,33)(27,39)(29,43)(31,37)(32,40)(35,36)(38,44)(41,45)(42,48)(46,47);(10,13)(14,16)(17,19,21,27)(18,24,23,36)(20,29,31,42)(22,35,33,30)(25,38,37,47)(26,39,34,28)(32,46,45,44)(40,43,41,48);(2,6)(5,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,31)(22,32)(23,37)(24,38)(26,41)(27,42)(28,43)(30,44)(33,45)(34,40)(35,46)(36,47)(39,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(20,31)(22,33)(23,34)(24,39)(25,40)(28,36)(29,42)(30,35)(37,41)(38,43)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6625653);