#8096825

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004021);