#8425356

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,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, 8332552);