#7857898

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

4597 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7765094);