#6718034

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,24,31,40)(20,33,36,25)(21,26,23,28)(29,45,46,38)(30,39,32,41)(34,48,47,42)(35,44,37,43);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,32)(22,31)(23,30)(25,42)(26,41)(27,40)(28,39)(29,35)(33,48)(36,47)(37,46)(38,43)(44,45);(10,15)(14,16)(17,20,22,36)(18,25,27,33)(19,29,31,46)(21,35,23,37)(24,38,40,45)(26,43,28,44)(30,34,32,47)(39,42,41,48);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,30,31,32)(20,35,36,37)(24,39,40,41)(25,43,33,44)(29,34,46,47)(38,42,45,48);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,30)(21,23)(24,39)(26,28)(29,34)(31,32)(35,37)(38,42)(40,41)(43,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6625230);