#8931060

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

4698 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=i,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, 8838256);