#7791052

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

4386 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=g,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, 7698248);