#8931421

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

4454 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,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, 8838617);