#6489414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,27,22,29,21,28)(19,30,32,24,34,44,33,41)(20,26,36,42,38,25,37,43)(31,46,48,40,35,45,47,39);(2,6)(7,8)(11,12)(14,15)(17,19)(18,24)(20,35)(21,32)(22,34)(23,33)(25,40)(26,39)(27,30)(28,44)(29,41)(31,38)(36,48)(37,47)(42,45)(43,46);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,31)(21,37)(22,38)(23,36)(24,39)(26,29)(27,43)(28,42)(30,45)(32,48)(33,47)(34,35)(40,41)(44,46);(9,12,13,11)(10,15,16,14)(17,20)(18,26)(19,31)(21,37)(22,38)(23,36)(24,40)(25,29)(27,42)(28,43)(30,46)(32,48)(33,47)(34,35)(39,41)(44,45);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,21,22,23)(18,27,29,28)(19,32,34,33)(20,36,38,37)(24,30,41,44)(25,42,26,43)(31,47,35,48)(39,46,40,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396610);