#7541794

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1233 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=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, 7448990);