#10460718

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

4069 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,e^2=h,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, 10367914);