#7824452

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

4419 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27,30,34)(26,31,33,38)(28,35,32,39)(29,36,37,40);(1,2,5,4)(3,6,8,7)(10,16)(15,18)(17,20)(23,24)(27,34)(31,38)(35,39)(36,40);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7731648);