#8075262

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4447 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)(11,18)(12,17)(13,19)(14,16)(15,21)(20,23)(22,24)(25,26)(27,29)(28,35)(30,32)(31,33)(34,37)(36,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,17)(14,21)(16,18)(19,22)(23,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,40)(38,39);(10,16)(13,23)(15,18)(17,24)(25,28,31,29)(26,27,33,35)(30,37,38,36)(32,39,40,34);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,17,16,24)(11,20,21,19)(13,18,23,15)(25,29,31,28)(26,27,33,35)(30,36,38,37)(32,39,40,34);(9,11)(10,18)(12,19)(13,17)(14,21)(15,16)(20,22)(23,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,40)(38,39);(9,13)(10,19)(11,17)(12,18)(14,23)(15,22)(16,20)(21,24)(25,30)(26,32)(27,34)(28,36)(29,37)(31,38)(33,40)(35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982458);