#7918522

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

4409 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)(10,14)(11,17)(16,22)(18,19)(25,26,30,34)(27,31,37,40)(28,32,36,39)(29,33,35,38);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,38)(33,39)(34,40);(9,10,13,14)(11,15,17,21)(12,16,20,22)(18,23,19,24)(25,28)(26,32)(27,29)(30,36)(31,33)(34,39)(35,37)(38,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,29,30,35)(26,33,34,38)(27,36,37,28)(31,39,40,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gh,e^2=g,f^2=ghi,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, 7825718);