#7609618

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

1766 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)(19,24)(20,25)(21,28)(22,27)(23,31)(26,37)(29,40)(30,39)(32,41)(33,48)(34,45)(35,44)(36,43)(38,46)(42,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,27)(25,42)(26,39)(28,41)(29,36)(32,37)(34,46)(35,47)(38,43)(40,45)(44,48);(10,14)(11,15)(17,19)(18,24)(20,29)(21,32)(22,31)(23,27)(25,40)(26,39)(28,41)(30,37)(33,36)(34,47)(35,46)(38,44)(42,45)(43,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,21,36,22,35,37,34)(18,25,28,43,27,44,26,45)(19,29,32,33,31,46,30,47)(23,38,39,42,24,40,41,48);(2,7)(6,8)(17,21,22,37)(18,26,27,28)(19,30,31,32)(20,34,35,36)(23,39,24,41)(25,43,44,45)(29,33,46,47)(38,48,40,42)

Polycyclic presentation:

2^9|b^2=fi,d^2=fhi,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, 7516814);