#7917804

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7825000);