#8077426

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	#47854	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26)(27,37)(28,40)(29,34)(30,38)(31,35)(32,36)(33,39);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,32,39)(26,33,36,37)(28,31,29,38)(30,34,35,40);(11,13)(16,17)(19,21)(23,24)(25,28,32,29)(26,34,36,40)(27,38,39,31)(30,37,35,33);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,16,13,17)(14,18,22,20)(19,24,21,23)(25,29,32,28)(26,34,36,40)(27,31,39,38)(30,37,35,33);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,30,32,35)(26,31,36,38)(27,34,39,40)(28,33,29,37);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,31)(26,35)(27,29)(28,39)(30,36)(32,38)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,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, 7984622);