#8096871

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004067);