#7025484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6932680);