#7842437

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

4567 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)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,24,22)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,21)(10,16,19,20)(12,14,17,24)(13,22,23,18)(25,27,31,29)(26,32,30,34)(28,35,38,36)(33,39,37,40);(11,21)(14,24)(16,20)(18,22)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,12)(10,13)(11,14)(15,17)(16,22)(18,20)(19,23)(21,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,13)(10,17)(11,22)(12,19)(14,20)(15,23)(16,24)(18,21)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,3,4,7)(2,6,5,8)(9,14)(10,18)(11,12)(13,20)(15,24)(16,23)(17,21)(19,22)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=h,f^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, 7749633);