#7506465

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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 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)(20,33)(21,36)(22,27)(23,26)(25,35)(28,45)(29,47)(30,44)(31,41)(32,40)(34,42)(37,43)(38,46)(39,48);(1,2,4,5)(3,8,7,6)(9,11,12,15)(10,13,14,16)(17,19,22,31)(18,24,27,41)(20,29,35,48)(21,32,23,30)(25,39,33,47)(26,44,36,40)(28,37,46,34)(38,42,45,43);(10,14)(13,16)(17,20)(18,25)(19,28)(21,34)(22,35)(23,37)(24,38)(26,42)(27,33)(29,30)(31,46)(32,48)(36,43)(39,44)(40,47)(41,45);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,39)(26,43)(27,33)(28,30)(31,48)(32,46)(36,42)(38,44)(40,45)(41,47);(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,40,41,44)(25,42,33,43)(28,29,46,48)(38,47,45,39)

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,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, 7413661);