#9851214

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
prime radical	#59017	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4274 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9758410);