#7609094

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,33)(21,27)(22,28)(23,29)(24,30)(26,35)(31,39)(32,40)(34,42)(36,43)(37,44)(38,45)(41,47)(46,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,25,29,40)(20,24,35,22)(21,34,37,47)(26,28,33,30)(27,42,44,41)(31,36,46,38)(39,43,48,45);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(17,20)(18,26)(19,22)(21,34)(23,35)(24,32)(25,30)(27,41)(28,40)(29,33)(31,36)(37,47)(38,46)(39,45)(42,44)(43,48);(10,14)(13,16)(17,21,24,38,23,37,22,36)(18,27,30,45,29,44,28,43)(19,31,20,34,32,46,35,47)(25,39,33,42,40,48,26,41);(2,7)(6,8)(17,22,23,24)(18,28,29,30)(19,20,32,35)(21,36,37,38)(25,33,40,26)(27,43,44,45)(31,34,46,47)(39,42,48,41)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7516290);