#6495108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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)(4,5)(6,8)(11,15)(14,16)(17,18)(19,28)(20,25)(21,36)(22,26)(23,31)(24,34)(27,29)(30,43)(32,44)(33,42)(35,40)(37,48)(38,47)(39,45)(41,46);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,29,45,22,32,21,31)(18,23,36,44,26,39,27,28)(20,33,46,47,34,30,35,48)(24,38,41,42,25,37,40,43);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(17,20)(18,24)(19,30)(21,35)(22,34)(23,37)(25,26)(27,40)(28,42)(29,46)(31,47)(32,33)(36,41)(38,39)(43,44)(45,48);(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,38)(24,26)(27,41)(28,43)(29,46)(31,47)(32,33)(36,40)(37,39)(42,44)(45,48);(9,11,12,15)(10,14,13,16)(18,26)(20,34)(23,39)(27,36)(28,44)(30,33)(35,46)(47,48)

Polycyclic presentation:

2^9|b^2=fg,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402304);