#7719282

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	#35071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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)(13,21)(16,20)(18,24)(22,23)(25,26,31,28)(27,34,33,30)(29,36,38,32)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27)(26,30)(28,34)(29,37)(31,33)(32,40)(35,38)(36,39);(10,12)(15,17)(16,20)(22,23)(26,28)(27,33)(29,38)(39,40);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,37,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,37)(34,40);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,34)(26,33,28,27)(29,39,38,40)(32,37,36,35)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=hi,f^2=g,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, 7626478);