#7732832

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

4391 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,30,32)(27,33,34,31)(28,37,38,29)(35,39,40,36);(1,3,5,8)(2,6,4,7)(25,27)(26,31)(28,35)(29,39)(30,34)(32,33)(36,37)(38,40);(10,11)(15,17)(16,18)(22,23)(26,32)(27,34)(28,38)(36,39);(1,4,5,2)(3,7,8,6)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,33)(28,35)(29,36)(30,34)(31,32)(37,39)(38,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28)(26,29)(27,35)(30,38)(31,39)(32,37)(33,36)(34,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,28)(27,36)(30,37)(31,40)(32,38)(33,35)(34,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=gh,e^2=hi,f^2=hi,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, 7640028);