#7743819

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
prime radical	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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)(11,15)(12,17)(13,18)(14,19)(16,20)(21,22)(23,24)(25,26,31,32)(27,29,30,33)(28,36,39,34)(35,40,38,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(26,32)(27,30)(34,36)(35,38);(1,3,4,7)(2,6,5,8)(11,20)(12,22)(15,16)(17,21)(25,27,31,30)(26,33,32,29)(28,35,39,38)(34,40,36,37);(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,28)(26,34)(27,35)(29,37)(30,38)(31,39)(32,36)(33,40);(9,12)(10,17)(11,18)(13,15)(14,22)(16,23)(19,21)(20,24)(25,29,31,33)(26,27,32,30)(28,37,39,40)(34,35,36,38);(9,13,14,23)(10,18,19,24)(11,21,20,17)(12,15,22,16)(25,30,31,27)(26,33,32,29)(28,38,39,35)(34,40,36,37)

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651015);