#7508611

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,30,23)(20,24,33,40)(22,26,35,42)(28,38,31,36)(29,39,32,37)(34,41,45,48)(43,47,44,46);(1,3)(2,6)(4,7)(5,8)(17,19,21,30)(18,23,25,27)(20,32,33,29)(22,28,35,31)(24,39,40,37)(26,36,42,38)(34,44,45,43)(41,47,48,46);(9,11,12,16)(10,13,14,15)(19,28)(22,35)(23,36)(26,42)(27,38)(29,43)(30,31)(32,44)(34,45)(37,46)(39,47)(41,48);(2,7)(3,8)(10,14)(13,15)(17,20)(18,24)(19,29)(21,33)(22,34)(23,37)(25,40)(26,41)(27,39)(28,43)(30,32)(31,44)(35,45)(36,46)(38,47)(42,48);(1,4,5,6)(2,3,7,8)(18,25)(20,33)(23,27)(26,42)(29,32)(34,45)(36,38)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7415807);