#8425455

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

4239 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)(3,10)(9,15)(11,13)(18,20)(19,25)(24,28)(27,29)(33,34,39,45)(35,40,37,42)(36,44,47,41)(38,46,48,43);(1,2,6,7)(3,8,10,4)(5,9,14,15)(11,16,13,12)(17,18,22,20)(19,26,25,23)(21,24,30,28)(27,32,29,31)(33,35)(34,40)(36,46)(37,39)(38,41)(42,45)(43,47)(44,48);(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,19,22,25)(18,23,20,26)(21,29,30,27)(24,32,28,31)(33,37,39,35)(34,42,45,40)(36,46,47,43)(38,44,48,41);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,38)(34,43)(35,41)(36,42)(37,44)(39,48)(40,47)(45,46);(1,5,6,14)(2,9,7,15)(3,11,10,13)(4,12,8,16)(17,21,22,30)(18,24,20,28)(19,27,25,29)(23,31,26,32)(33,36)(34,44)(35,46)(37,43)(38,42)(39,47)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

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, 8332651);