#10480845

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^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, 10388041);