#7620467

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1235 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)(19,24)(20,32)(21,38)(22,27)(23,28)(25,36)(26,31)(29,42)(30,41)(33,40)(34,43)(35,45)(37,44)(39,46)(47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,41)(20,33,36,29)(21,39,31,22)(25,42,32,40)(26,27,38,46)(34,37,47,35)(43,44,48,45);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,36)(24,40)(26,43)(27,45)(28,32)(30,33)(31,47)(35,39)(38,48)(41,42)(44,46);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,22,19,23,31,39,30)(18,26,27,41,28,38,46,24)(20,34,37,29,36,47,35,33)(25,43,45,42,32,48,44,40);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,22,23,39)(18,27,28,46)(19,21,30,31)(20,35,36,37)(24,38,41,26)(25,44,32,45)(29,47,33,34)(40,43,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fghi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527663);