#7954571

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	#47927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,39)(35,36)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,3,5,4)(2,7,6,8)(10,16)(12,15)(17,23)(20,22)(26,28)(27,35)(33,37)(34,38);(1,3,5,4)(2,7,6,8)(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,28,31,26)(27,36,35,32)(29,37,39,33)(30,38,40,34);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,29)(26,33)(27,34)(28,37)(30,36)(31,39)(32,40)(35,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,30)(26,34)(27,37)(28,38)(29,32)(31,40)(33,35)(36,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gi,f^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, 7861767);