#6455823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,28,32,24)(20,33,36,25)(21,37,23,26)(29,39,47,45)(30,41,31,40)(34,42,46,48)(35,43,38,44);(2,5)(6,8)(17,19)(18,24)(20,34)(21,31)(22,32)(23,30)(25,42)(26,40)(27,28)(29,35)(33,48)(36,46)(37,41)(38,47)(39,44)(43,45);(10,13)(14,16)(17,20,22,36)(18,25,27,33)(19,29,32,47)(21,35,23,38)(24,39,28,45)(26,43,37,44)(30,34,31,46)(40,42,41,48);(1,3,4,7)(2,6,5,8)(19,30)(21,23)(24,40)(26,37)(28,41)(29,34)(31,32)(35,38)(39,42)(43,44)(45,48)(46,47);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,37)(19,31,32,30)(20,35,36,38)(24,40,28,41)(25,43,33,44)(29,46,47,34)(39,42,45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363019);