#10368455

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

4237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275651);