#8095000

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

4221 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,21)(19,25)(24,29)(26,31)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(2,6)(7,13)(8,11)(14,16)(20,28)(24,29)(26,31)(30,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=gi,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, 8002196);