#7918717

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	#35681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4719 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825913);