#8228234

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

4190 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:

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

Polycyclic presentation:

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

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, 8135430);