#6458250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1922 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:

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

Polycyclic presentation:

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

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, 6365446);