#10481031

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,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, 10388227);