#7728227

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

4364 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)(9,10,15,17)(11,20,12,16)(13,14,19,18)(21,22,23,24);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,17,20)(13,21,19,23)(14,24,18,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,5,4)(2,7,6,8)(10,17)(11,12)(13,19)(22,24)(26,31)(28,32)(34,37)(36,39);(9,12,15,11)(10,16,17,20)(13,23,19,21)(14,24,18,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,13,15,19)(10,18,17,14)(11,21,12,23)(16,22,20,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,19,17,13)(11,22,12,24)(16,21,20,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7635423);