#7576243

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=fi,e^2=gi,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, 7483439);