#7954642

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

4485 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=ghi,e^2=g,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, 7861838);