#10481009

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

4002 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)(5,13)(6,7)(8,11)(9,15)(10,16)(12,14)(20,28)(23,29)(26,32)(30,31)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,48)(46,47);(2,7)(4,8)(9,13)(12,16)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,48,41,46);(1,3)(2,8)(4,7)(5,10)(6,11)(9,16)(12,13)(14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,46,44,48);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,45,44,47);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,46)(47,48);(1,5,6,15)(2,9,7,13)(3,10,11,14)(4,12,8,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gi,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, 10388205);