#6469300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1791 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18)(19,28)(20,25)(21,35)(22,27)(23,30)(24,33)(26,36)(29,43)(31,44)(32,42)(34,40)(37,48)(38,46)(39,47)(41,45);(9,10)(11,16)(12,14)(13,15)(17,19,21,30,22,31,36,47)(18,23,26,28,27,39,35,44)(20,32,34,48,33,29,45,46)(24,38,40,43,25,37,41,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,27)(26,40)(28,42)(30,46)(31,32)(35,41)(36,45)(38,39)(43,44)(47,48);(10,14)(13,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,38)(24,27)(26,41)(28,43)(30,46)(31,32)(35,40)(36,45)(37,39)(42,44)(47,48);(1,4,5,3)(2,6,7,8)(20,33)(24,25)(29,32)(34,45)(37,38)(40,41)(42,43)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376496);