#7609626

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,22,27)(29,37,46,48)(30,38,35,43)(32,40,45,47)(33,41,36,44);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,30)(22,35)(23,31)(25,40)(26,38)(27,43)(28,39)(29,33)(34,45)(36,46)(37,41)(42,47)(44,48);(10,14)(11,15)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,43)(27,38)(28,39)(32,33)(34,46)(36,45)(40,41)(42,48)(44,47);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,33,23,34,21,36)(18,25,27,41,28,42,26,44)(19,29,30,32,31,46,35,45)(24,37,38,40,39,48,43,47);(2,7)(6,8)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fh,e^2=i,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, 7516822);