#7870500

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

4516 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)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26)(27,35)(28,36)(29,33)(30,34)(31,37)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(25,27,30,37)(26,31,34,35)(28,33,38,40)(29,36,39,32);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,26,30,34)(27,35,37,31)(28,32,38,36)(29,33,39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(1,2,5,4)(3,6,8,7)(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,28)(26,32)(27,33)(29,31)(30,38)(34,36)(35,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,29)(26,33)(27,36)(28,35)(30,39)(31,38)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=gh,f^2=gh,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, 7777696);