#8087854

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 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)(9,10,14,13)(11,15,19,18)(12,16,22,21)(17,23,20,24)(25,26,31,35)(27,33,29,32)(28,30,34,40)(36,37,38,39);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,32,35,33)(28,38,34,36)(30,37,40,39);(10,13)(15,18)(16,21)(23,24)(27,29)(32,33)(36,38)(37,39);(9,11,14,19)(10,15,13,18)(12,20,22,17)(16,24,21,23)(25,28)(26,30)(27,36)(29,38)(31,34)(32,39)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,12,14,22)(10,16,13,21)(11,17,19,20)(15,23,18,24)(25,29,31,27)(26,33,35,32)(28,36,34,38)(30,39,40,37);(9,13,14,10)(11,18,19,15)(12,21,22,16)(17,24,20,23)(25,30)(26,34)(27,37)(28,35)(29,39)(31,40)(32,36)(33,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=h,e^2=gh,f^2=h,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, 7995050);