#7710902

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	#37452	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4648 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)(12,21)(13,22)(14,16)(17,20)(18,23)(19,24)(28,36)(32,37)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,19)(10,15,16,24)(12,13,20,23)(17,18,21,22)(26,30)(31,33)(32,37)(38,39);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,20)(18,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,21)(22,23);(9,12,14,20)(10,17,16,21)(11,13,19,23)(15,18,24,22)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,18)(11,20)(12,19)(14,23)(15,21)(16,22)(17,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618098);