#7787761

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

4474 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)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,27,30,29)(26,31,33,34)(28,35,38,37)(32,39,36,40);(10,15)(11,17)(13,21)(20,23)(27,29)(31,34)(35,37)(39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,26)(27,34)(28,36)(29,31)(30,33)(32,38)(35,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,30,27)(26,31,33,34)(28,37,38,35)(32,39,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7694957);