#7620533

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1232 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgh,e^2=g,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, 7527729);