#7989829

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

4674 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)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26)(27,30)(28,36)(29,31)(32,34)(33,38)(35,40)(37,39);(1,3,5,8)(2,6,4,7)(10,15)(12,19)(17,21)(20,24)(25,27,29,34)(26,30,31,32)(28,37,38,35)(33,40,36,39);(10,15)(11,16)(17,21)(18,23)(26,31)(27,34)(33,36)(35,37);(1,4,5,2)(3,7,8,6)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,27)(26,32)(28,35)(29,34)(30,31)(33,40)(36,39)(37,38);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,26)(27,32)(28,33)(29,31)(30,34)(35,40)(36,38)(37,39);(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,28)(26,33)(27,35)(29,38)(30,39)(31,36)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7897025);