#10040596

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3795 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9947792);