#8504823

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

4101 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=i,e^2=gh,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, 8412019);