#10534334

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

4076 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18)(19,23)(20,25)(21,31)(22,27)(24,30)(26,28)(29,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,8)(2,5,10,4)(6,11,15,16)(9,13,14,12)(17,19,22,29)(18,23,27,32)(20,28,30,21)(24,31,25,26)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,27)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,5)(4,8)(6,14)(7,10)(9,15)(11,12)(13,16)(17,18)(19,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,21,22,28)(18,26,27,31)(19,20,29,30)(23,24,32,25)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=g,f^2=ghi,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, 10441530);