#7505369

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,d^2=f,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, 7412565);