#6488684

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6395880);