#9851068

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=g,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, 9758264);