#7618130

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(10,14)(11,12)(17,18,22,26)(19,23,28,35)(20,30,32,24)(21,25,33,38)(27,39,41,34)(29,36,42,46)(31,43,44,37)(40,47,48,45);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(20,31)(21,33)(24,37)(25,38)(27,40)(29,42)(30,43)(32,44)(34,45)(36,46)(39,47)(41,48);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,34)(25,36)(26,35)(30,39)(31,40)(32,41)(33,42)(37,45)(38,46)(43,47)(44,48);(9,11,13,12)(10,15,14,16)(17,20,21,31,22,32,33,44)(18,24,25,37,26,30,38,43)(19,27,29,40,28,41,42,48)(23,34,36,45,35,39,46,47);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(19,28)(20,32)(23,35)(24,30)(29,42)(31,44)(36,46)(37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7525326);