#9863912

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

4027 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9771108);