#6635582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632068
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	#23924	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,21,26,22,27,34,24)(19,23,30,38,29,37,44,35)(20,25,33,42,32,41,46,39)(28,36,43,48,31,40,45,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,29)(18,23,27,37)(20,31,32,28)(21,30,34,44)(24,35,26,38)(25,40,41,36)(33,45,46,43)(39,48,42,47);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(18,24)(21,34)(23,35)(25,39)(26,27)(30,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,32)(18,25,27,41)(19,28,29,31)(21,33,34,46)(23,36,37,40)(24,39,26,42)(30,43,44,45)(35,47,38,48);(2,6)(5,8)(19,29)(20,32)(23,37)(25,41)(30,44)(33,46)(35,38)(39,42)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,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, 6542778);