#7616013

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1360 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)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,28,23)(20,31,35,24)(22,26,34,40)(29,36,33,39)(30,41,42,37)(32,45,46,38)(43,47,44,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,34)(23,27)(24,38)(26,40)(29,43)(31,45)(33,44)(35,46)(36,47)(39,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,23)(20,33)(21,28)(22,30)(24,39)(25,27)(26,37)(29,35)(31,36)(32,43)(34,42)(38,47)(40,41)(44,46)(45,48);(10,14)(13,15)(17,20,34,46,21,35,22,32)(18,24,40,45,25,31,26,38)(19,29,42,43,28,33,30,44)(23,36,41,47,27,39,37,48);(2,5)(6,8)(10,14)(13,15)(19,28)(20,35)(23,27)(24,31)(30,42)(32,46)(37,41)(38,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7523209);