#8095077

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

4133 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002273);