#8110595

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4212 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,18,13,16)(12,20,21,17)(19,23,22,24)(25,26)(27,31)(28,37)(29,39)(30,34)(32,38)(33,35)(36,40);(2,3)(6,7)(10,15)(12,21)(16,18)(19,22)(25,26,30,34)(27,31,36,40)(28,32,38,37)(29,33,35,39);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,16,15,18)(12,19,21,22)(17,23,20,24)(25,27,30,36)(26,31,34,40)(28,35,38,29)(32,39,37,33);(1,3,5,2)(4,7,8,6)(9,12,14,21)(10,17,15,20)(11,19,13,22)(16,23,18,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,40)(34,37);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(9,13,14,11)(10,18,15,16)(12,22,21,19)(17,24,20,23)(25,29,30,35)(26,33,34,39)(27,28,36,38)(31,32,40,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^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, 8017791);