#7510509

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7417705);