#7773928

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7681124);