#7728244

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7635440);