#7727020

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

4628 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=hi,f^2=h,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, 7634216);