#7815813

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

4442 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=h,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, 7723009);