#9851200

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

4112 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,25)(19,21,27,23)(20,24,29,32)(26,30,31,28)(34,40)(36,44)(41,46)(43,45);(3,9)(7,13)(10,15)(14,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,34,39,40)(35,41,37,46)(36,42,44,47)(38,43,48,45);(2,6)(7,13)(8,11)(14,16)(17,19,22,27)(18,23,25,21)(20,26,29,31)(24,28,32,30)(33,35,39,37)(34,41,40,46)(36,43,44,45)(38,42,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31)(33,36,39,44)(34,42,40,47)(35,43,37,45)(38,41,48,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,24)(19,26)(21,30)(22,29)(23,28)(25,32)(27,31)(33,37,39,35)(34,41,40,46)(36,45,44,43)(38,42,48,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,28)(22,23)(24,31)(25,27)(26,32)(29,30)(33,38,39,48)(34,43,40,45)(35,42,37,47)(36,41,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9758396);