#7620463

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

1239 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)(22,40,24,28)(31,42,35,43)(36,44,47,48)(37,45,41,46);(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)(26,43,34,42)(27,40,39,28)(36,37,47,41)(44,46,48,45);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,36)(22,41)(23,38)(24,37)(25,42)(27,44)(28,46)(29,34)(30,43)(32,47)(33,35)(39,48)(40,45);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,21,24,19,23,32,22,33)(18,27,28,25,29,39,40,30)(20,36,37,31,38,47,41,35)(26,44,46,42,34,48,45,43);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,40)(19,32,33,21)(20,37,38,41)(25,27,30,39)(26,45,34,46)(31,36,35,47)(42,48,43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,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,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, 7527659);