#7888601

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

4506 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=i,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, 7795797);