#7513501

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,31,35,25)(20,27,21,26)(22,28,24,30)(32,40,48,47)(33,41,37,44)(34,46,39,42)(36,45,38,43);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,23,35)(18,25,29,31)(20,33,21,37)(22,34,24,39)(26,41,27,44)(28,42,30,46)(32,38,48,36)(40,45,47,43);(10,13)(14,16)(17,20,23,21)(18,26,29,27)(19,32,35,48)(22,36,24,38)(25,40,31,47)(28,43,30,45)(33,34,37,39)(41,42,44,46);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,26)(19,33,35,37)(22,36,24,38)(25,41,31,44)(28,43,30,45)(32,34,48,39)(40,42,47,46);(1,5,6,4)(2,3,7,8)(17,22,23,24)(18,28,29,30)(19,34,35,39)(20,36,21,38)(25,42,31,46)(26,43,27,45)(32,37,48,33)(40,44,47,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=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, 7420697);