#7506445

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,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, 7413641);