#7510149

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16)(17,18)(19,28)(20,25)(21,36)(22,27)(23,26)(24,31)(29,45)(30,39)(32,44)(33,48)(34,43)(35,42)(37,41)(38,46)(40,47);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,33)(21,32)(22,31)(23,30)(25,40)(26,44)(27,28)(29,34)(35,47)(36,39)(37,46)(38,43)(41,45)(42,48);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,41)(27,42)(28,45)(30,33)(31,46)(32,47)(36,43)(39,48)(40,44);(2,7)(3,8)(10,15)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,44)(25,41,42,43)(29,33,46,47)(38,48,45,40);(1,4,5,6)(2,3,7,8)(18,27)(19,31)(25,42)(26,36)(29,46)(30,32)(33,47)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

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, 7417345);