#6489967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(19,29)(20,26)(21,27)(22,28)(23,25)(24,32)(30,43)(31,37)(33,39)(34,40)(35,41)(36,42)(38,45)(44,48)(46,47);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19,22,32)(18,24,28,29)(20,30,35,45)(21,31,23,33)(25,37,27,39)(26,38,41,43)(34,44,36,46)(40,47,42,48);(9,12,14,13)(10,15,11,16)(18,25)(21,23)(24,37)(26,40)(27,28)(29,39)(31,33)(34,36)(38,47)(41,42)(43,48)(44,46);(9,12,14,13)(10,15,11,16)(17,20,22,35)(18,26,28,41)(19,30,32,45)(21,36,23,34)(24,38,29,43)(25,40,27,42)(31,46,33,44)(37,47,39,48);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,27,28,25)(19,31,32,33)(20,34,35,36)(24,39,29,37)(26,40,41,42)(30,44,45,46)(38,47,43,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi,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, 6397163);