#7798103

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

4500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7705299);