#8932195

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	#45740	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,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, 8839391);