#7582307

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,26)(21,27)(22,38)(23,28)(24,29)(30,39)(31,41)(32,42)(33,40)(34,43)(35,46)(36,44)(37,45)(47,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,20)(18,25,29,26)(21,34,37,47)(22,31,23,33)(27,43,45,48)(28,40,38,41)(30,35,32,36)(39,46,42,44);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,24,19)(18,26,29,25)(21,32,37,30)(22,31,23,33)(27,42,45,39)(28,40,38,41)(34,36,47,35)(43,44,48,46);(10,14)(13,15)(17,21,24,37)(18,27,29,45)(19,30,20,32)(22,36,23,35)(25,39,26,42)(28,46,38,44)(31,47,33,34)(40,43,41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,20,33)(21,35,37,36)(25,40,26,41)(27,44,45,46)(30,34,32,47)(39,48,42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489503);