#7569694

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,28,23)(20,31,34,24)(22,26,35,41)(29,37,33,40)(30,43,44,38)(32,42,36,39)(45,47,46,48);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,28)(20,32)(22,35)(23,27)(24,39)(26,41)(29,45)(31,42)(33,46)(34,36)(37,47)(40,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,33)(21,28)(22,30)(24,40)(25,27)(26,38)(29,34)(31,37)(32,45)(35,44)(36,46)(39,47)(41,43)(42,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,24)(19,29)(21,34)(22,36)(23,37)(25,31)(26,42)(27,40)(28,33)(30,45)(32,35)(38,47)(39,41)(43,48)(44,46);(2,6)(7,8)(10,14)(13,15)(19,28)(20,34)(23,27)(24,31)(30,44)(32,36)(38,43)(39,42)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=fgi,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,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7476890);