#6908060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815256);