#8492950

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

4156 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)(3,9)(4,11)(5,8)(6,7)(10,16)(12,13)(14,15)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(2,7)(4,9)(8,15)(12,16)(19,28)(21,29)(24,32)(25,30)(33,34)(35,43)(36,47)(37,48)(38,42)(39,46)(40,45)(41,44);(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,3)(2,4)(5,13)(6,11)(7,9)(8,16)(10,14)(12,15)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=h,e^2=i,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, 8400146);