#7505389

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(19,28)(20,26)(21,34)(22,24)(23,27)(25,31)(29,38)(30,43)(32,37)(33,42)(35,41)(36,40)(39,46)(44,47)(45,48);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,13,14,16)(18,24)(19,29)(20,32)(21,23)(25,38)(28,40)(30,45)(31,36)(39,48)(41,42)(43,47)(44,46);(1,4,6,5)(2,3,7,8)(10,14)(13,16)(17,19)(18,25)(20,30)(21,29)(22,31)(23,36)(24,28)(26,39)(27,38)(32,46)(33,45)(34,40)(35,44)(37,43)(41,48)(42,47);(1,5,6,4)(2,3,7,8)(10,14)(13,16)(17,20,22,32)(18,26,24,37)(19,30,31,46)(21,35,23,33)(25,39,28,43)(27,42,34,41)(29,44,36,45)(38,47,40,48);(17,21,22,23)(18,27,24,34)(19,29,31,36)(20,33,32,35)(25,38,28,40)(26,41,37,42)(30,45,46,44)(39,48,43,47)

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,e^b=ei,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, 7412585);