#7806833

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

4793 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)(9,10,13,17)(11,16,18,14)(12,15,21,24)(19,23,20,22)(25,26,30,31)(27,35,36,32)(28,38,39,33)(29,34,40,37);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,14,17,16)(12,20,21,19)(15,23,24,22)(26,31)(27,36)(28,39)(29,40);(11,18)(14,16)(19,20)(22,23)(25,27,30,36)(26,32,31,35)(28,37,39,34)(29,38,40,33);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,27)(26,32)(28,37)(29,33)(30,36)(31,35)(34,39)(38,40);(1,4,5,2)(3,7,8,6)(9,12,13,21)(10,15,17,24)(11,19,18,20)(14,22,16,23)(25,28,30,39)(26,33,31,38)(27,37,36,34)(29,35,40,32);(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,21)(22,24)(25,29)(26,34)(27,33)(28,35)(30,40)(31,37)(32,39)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7714029);