#8930319

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

4705 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=ghi,e^2=h,f^2=g,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, 8837515);