#8533188

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

4128 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,3)(2,9)(4,14)(5,11)(6,10)(7,15)(8,16)(12,13)(19,25)(22,28)(26,31)(29,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,4,8,5)(2,10,12,7)(3,14,16,11)(6,13,15,9)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5,8,4)(2,10,12,7)(3,11,16,14)(6,13,15,9)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,6,8,15)(2,11,12,14)(3,10,16,7)(4,9,5,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,7,8,10)(2,5,12,4)(3,15,16,6)(9,11,13,14)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440384);