#8137386

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

4151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044582);