#7616668

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1259 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)(3,7)(4,8)(5,6)(11,12)(14,15)(17,18,21,23)(19,27,28,24)(20,31,34,25)(22,26,35,36)(29,37,33,40)(30,42,43,38)(32,39,46,41)(44,48,45,47);(2,6)(7,8)(9,10,13,16)(11,15,12,14)(18,23)(19,28)(20,32)(22,35)(25,39)(29,44)(31,41)(33,45)(34,46)(37,47)(38,42)(40,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,19,21,28)(18,24,23,27)(20,33,34,29)(22,30,35,43)(25,40,31,37)(26,38,36,42)(32,44,46,45)(39,47,41,48);(9,12,13,11)(10,15,16,14)(17,20,22,32,21,34,35,46)(18,25,26,41,23,31,36,39)(19,29,30,45,28,33,43,44)(24,37,38,47,27,40,42,48);(1,4,5,3)(2,7,6,8)(18,23)(19,28)(25,31)(26,36)(29,33)(30,43)(39,41)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523864);