#10320573

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

4653 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,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, 10227769);