#7783247

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

4408 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7690443);