#8504801

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

3941 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411997);