#7846826

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

4428 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)(9,10,14,13)(11,16,18,15)(12,20,22,17)(19,23,21,24)(25,26,31,29)(27,32,30,34)(28,33,37,38)(35,39,36,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,23)(20,24)(21,22)(25,27)(26,32)(28,35)(29,34)(30,31)(33,39)(36,37)(38,40);(10,13)(11,18)(17,20)(19,21)(27,30)(32,34)(35,36)(39,40);(9,11)(10,16)(12,21)(13,15)(14,18)(17,23)(19,22)(20,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,19)(13,20)(14,22)(15,23)(16,24)(18,21)(25,29,31,26)(27,34,30,32)(28,33,37,38)(35,39,36,40);(9,13,14,10)(11,15,18,16)(12,20,22,17)(19,23,21,24)(25,30,31,27)(26,34,29,32)(28,36,37,35)(33,40,38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=gi,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, 7754022);