#7972327

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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)(11,18)(17,21)(19,24)(25,26,30,32)(27,35,36,31)(28,39,38,33)(29,34,37,40);(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(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)(12,16)(17,21)(20,22)(26,32)(27,36)(28,38)(34,40);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,28)(26,33)(27,37)(29,36)(30,38)(31,40)(32,39)(34,35);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(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=g,c^2=gi,e^2=gh,f^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, 7879523);