#6579982

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487178);