#10320629

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

4597 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)(26,32)(27,28)(30,34)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,38)(39,40);(2,5)(6,7)(10,15)(11,17)(16,21)(18,22)(25,27,31,28)(26,33,32,36)(29,37,35,39)(30,38,34,40);(2,5)(6,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,40,34,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,40,36,38);(1,3)(2,6)(4,8)(5,7)(9,13)(10,16)(11,18)(12,19)(14,23)(15,21)(17,22)(20,24)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,39,36,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=g,f^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, 10227825);