#8077292

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	#46414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4644 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984488);