#10102228

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

4554 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)(3,4)(5,7)(6,8)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(26,31)(27,34)(29,37)(36,39);(1,3,6,5)(2,7,8,4)(10,15)(12,21)(16,19)(18,22)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(9,11,14,20)(10,16,15,19)(12,18,21,22)(13,17,23,24)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4,6,7)(2,5,8,3)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,5,6,3)(2,7,8,4)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,2)(3,7)(4,5)(6,8)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,29,30,37)(26,33,31,38)(27,35,34,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009424);