#8535232

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

4241 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,24,28)(19,27,22,23)(20,26,21,25)(29,30,31,32)(33,34,38,41)(35,42,37,39)(36,40,45,47)(43,48,44,46);(3,4)(7,10)(11,12)(15,16)(17,19,24,22)(18,23,28,27)(20,29,21,31)(25,30,26,32)(33,35,38,37)(34,39,41,42)(36,43,45,44)(40,46,47,48);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,20,24,21)(18,25,28,26)(19,29,22,31)(23,30,27,32)(33,36,38,45)(34,40,41,47)(35,43,37,44)(39,46,42,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,24,20)(18,26,28,25)(19,29,22,31)(23,30,27,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,48)(45,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,24,19)(18,27,28,23)(20,29,21,31)(25,30,26,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,23,24,27)(18,19,28,22)(20,30,21,32)(25,29,26,31)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=hi,f^2=g,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, 8442428);