#10247412

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

4199 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,22,23)(19,26,25,21)(20,24,28,32)(27,30,29,31)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(2,3)(7,9)(8,10)(15,16)(18,23)(19,25)(20,28)(30,31)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(18,23)(19,25)(24,32)(27,29)(34,39)(35,42)(40,45)(43,48);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,19,22,25)(18,21,23,26)(20,29,28,27)(24,30,32,31)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,29)(26,30)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,25)(19,23)(20,30)(22,26)(24,27)(28,31)(29,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154608);