#8096922

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

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=gi,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, 8004118);