#8932172

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

4268 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)(3,4)(5,6)(7,8)(9,10,15,18)(11,14,13,17)(12,21,23,16)(19,22,20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,37)(38,40);(2,6)(3,7)(9,11,15,13)(10,14,18,17)(12,20,23,19)(16,22,21,24)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3,5,7)(2,4,6,8)(9,12,15,23)(10,16,18,21)(11,19,13,20)(14,24,17,22)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(1,4)(2,7)(3,6)(5,8)(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,24)(18,21)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(9,13,15,11)(10,17,18,14)(12,20,23,19)(16,22,21,24)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(9,14,15,17)(10,11,18,13)(12,22,23,24)(16,20,21,19)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839368);