#10360946

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

4245 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268142);