#7602878

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1361 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,34)(21,40)(22,28)(23,29)(24,30)(26,38)(27,33)(31,43)(32,42)(35,41)(36,46)(37,45)(39,44)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,43)(27,30)(28,40)(29,42)(31,38)(34,41)(36,39)(37,47)(44,46)(45,48);(10,14)(11,16)(17,20,23,38,24,39,22,37)(18,26,28,34,30,45,29,44)(19,31,21,35,33,47,32,36)(25,41,42,43,27,46,40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,33,24,32,22,19)(18,27,28,40,30,25,29,42)(20,36,38,31,39,35,37,47)(26,41,34,43,45,46,44,48);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,21,33,32)(20,37,39,38)(25,40,27,42)(26,34,45,44)(31,35,47,36)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

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, 7510074);