#10464568

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

4291 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371764);