#10481048

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,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, 10388244);