#7989573

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(10,15)(11,17)(16,21)(18,24)(25,26,30,32)(27,35,36,31)(28,39,38,33)(29,34,37,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,21)(14,15)(16,23)(17,19)(18,22)(20,24)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,40,39,34);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(16,21)(20,22)(26,32)(27,36)(28,38)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,40)(38,39);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,22)(14,19)(15,17)(16,24)(18,21)(20,23)(25,28)(26,33)(27,37)(29,36)(30,38)(31,40)(32,39)(34,35);(9,13,14,23)(10,16,15,21)(11,18,17,24)(12,20,19,22)(25,29,30,37)(26,34,32,40)(27,38,36,28)(31,39,35,33)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=gi,e^2=gi,f^2=i,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, 7896769);