#8096856

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

4223 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=gh,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, 8004052);