#10455711

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

4157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 10362907);