#7830798

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

4649 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(10,15)(11,16)(13,22)(19,24)(27,33)(31,35)(34,39)(38,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40);(9,13)(10,17)(11,18)(12,19)(14,22)(15,21)(16,23)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,e^2=gi,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, 7737994);