#10444918

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

4749 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)(4,5)(6,8)(9,10)(11,17)(12,16)(13,15)(14,21)(18,22)(19,23)(20,24)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(2,5)(6,7)(10,13)(16,17)(18,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,13,17)(14,19,22,20)(18,23,21,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,17,13,16)(14,19,22,20)(18,23,21,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,13,15,10)(11,17,12,16)(14,21,22,18)(19,24,20,23)(25,28)(26,32)(27,34)(29,36)(30,31)(33,40)(35,37)(38,39);(9,14,15,22)(10,18,13,21)(11,19,12,20)(16,23,17,24)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10352114);