#8110599

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017795);