#7969653

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	#34829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4479 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)(11,14)(12,20)(13,17)(15,19)(16,22)(18,24)(21,23)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(26,31)(27,34)(33,36)(35,40);(9,11,13,19)(10,15,17,14)(12,21,22,18)(16,23,20,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,20)(19,21)(25,29)(26,27)(28,37)(30,32)(31,34)(33,35)(36,40)(38,39);(9,10)(11,15)(12,20)(13,17)(14,19)(16,22)(18,23)(21,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7876849);