#6698097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(4,5)(7,8)(10,14)(11,12)(17,18)(19,28)(20,33)(21,27)(22,25)(23,26)(24,32)(29,38)(30,39)(31,45)(34,44)(35,48)(36,41)(37,42)(40,47)(43,46);(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,30)(21,35)(22,32)(23,29)(25,28)(26,39)(27,43)(31,37)(33,38)(34,47)(36,46)(40,42)(41,48)(44,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(20,23)(24,38)(27,41)(28,39)(30,32)(31,46)(34,37)(35,47)(40,43)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,22,23)(18,26,25,33)(19,30,32,29)(21,34,36,37)(24,39,28,38)(27,42,41,44)(31,35,47,46)(40,48,45,43);(10,14)(15,16)(17,21,22,36)(18,27,25,41)(19,31,32,47)(20,34,23,37)(24,40,28,45)(26,42,33,44)(29,46,30,35)(38,43,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6605293);