#8425359

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

4196 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,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, 8332555);