#10463953

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

4415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=h,e^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, 10371149);