#8188110

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,19,23,16)(12,17,13,22)(14,18,24,21)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,24)(19,20)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(2,5)(6,7)(9,12,15,13)(10,17,20,22)(11,21,23,18)(14,16,24,19)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(9,13,15,12)(10,17,20,22)(11,21,23,18)(14,19,24,16)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,24)(10,18,20,21)(11,22,23,17)(12,16,13,19)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,19)(12,18)(13,21)(14,22)(15,23)(16,20)(17,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095306);