#6489104

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

1706 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)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,24,29)(19,30,34,41)(20,26,21,27)(22,28,23,37)(25,38,42,33)(31,45,32,46)(35,43,36,44)(39,47,40,48);(11,12)(14,15)(17,19)(18,25)(20,31)(21,32)(22,33)(23,38)(24,34)(26,39)(27,40)(28,41)(29,42)(30,37)(35,47)(36,48)(43,45)(44,46);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,26)(19,31)(21,24)(22,36)(23,35)(25,39)(27,29)(28,44)(30,45)(32,34)(33,48)(37,43)(38,47)(40,42)(41,46);(9,12,13,11)(10,15,16,14)(17,21,24,20)(18,27,29,26)(19,32,34,31)(22,35,23,36)(25,40,42,39)(28,43,37,44)(30,46,41,45)(33,47,38,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(17,22,24,23)(18,28,29,37)(19,33,34,38)(20,35,21,36)(25,41,42,30)(26,43,27,44)(31,47,32,48)(39,45,40,46)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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,d^c=di,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, 6396300);