#7715799

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	#36737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4720 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)(11,16)(12,14)(13,21)(15,17)(18,22)(19,23)(20,24)(26,31)(29,39)(32,40)(33,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,39)(37,40);(10,17)(11,12)(13,22)(23,24)(26,31)(28,38)(32,40)(34,35);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,21,24)(25,27,30,37)(26,32,31,40)(28,35,38,34)(29,36,39,33);(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,24)(17,21)(25,28)(26,33)(27,35)(29,32)(30,38)(31,36)(34,37)(39,40);(9,14)(10,11)(12,17)(13,23)(15,16)(18,19)(20,21)(22,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,35)(33,37)(38,40)

Polycyclic presentation:

2^9|b^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7622995);