#10322952

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,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, 10230148);