#7842429

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

4390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=i,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, 7749625);