#6616036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,31)(25,34)(29,44)(30,39)(32,43)(35,41)(36,40)(38,47)(42,45)(46,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,34,45,35,46)(25,38,40,44,33,48,41,42);(2,6)(7,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,41)(27,33)(28,42)(30,47)(31,46)(32,45)(37,40)(39,48)(43,44);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,34)(25,33)(29,45)(35,36)(38,48)(40,41)(42,44)(46,47);(1,4,5,3)(2,8,6,7)(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,43)(25,40,33,41)(29,46,45,47)(38,44,48,42)

Polycyclic presentation:

2^9|b^2=fgh,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

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, 6523232);