#7891197

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

4600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7798393);