#8908042

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

4575 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815238);