#6488774

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(12,15)(13,16)(17,18,24,39,23,29,22,28)(19,30,38,25,34,44,33,42)(20,26,37,27,35,43,21,36)(31,41,32,40,46,48,47,45);(11,12)(14,15)(17,19)(18,25)(20,31)(21,32)(22,38)(23,34)(24,33)(26,40)(27,41)(28,44)(29,42)(30,39)(35,46)(36,48)(37,47)(43,45);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,31)(21,22)(23,35)(24,37)(25,40)(27,39)(28,36)(29,43)(30,41)(32,38)(33,47)(34,46)(42,45)(44,48);(9,12,13,11)(10,15,16,14)(17,21,23,37)(18,27,29,36)(19,32,34,47)(20,22,35,24)(25,41,42,48)(26,39,43,28)(30,45,44,40)(31,38,46,33);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(20,35)(21,37)(26,43)(27,36)(31,46)(32,47)(40,45)(41,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6395970);