#7534562

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(11,15)(12,14)(13,16)(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);(2,6)(7,8)(11,16)(13,15)(17,19,24,34)(18,25,30,42)(20,35,21,31)(22,33,23,38)(26,43,27,39)(28,41,29,46)(32,36,47,37)(40,44,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(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);(10,14)(13,15)(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);(1,4,5,3)(2,7,6,8)(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=i,c^2=fgi,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, 7441758);