#7510501

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=f,d^2=fgi,e^2=i,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, 7417697);