#8188112

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

4339 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)(9,10,16,19)(11,13,21,12)(14,17,20,23)(15,18,24,22)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,6)(4,5)(7,8)(9,11,16,21)(10,12,19,13)(14,15,20,24)(17,22,23,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,5)(6,7)(9,12,16,13)(10,11,19,21)(14,22,20,18)(15,23,24,17)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(9,13,16,12)(10,11,19,21)(14,18,20,22)(15,23,24,17)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14,16,20)(10,17,19,23)(11,15,21,24)(12,22,13,18)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,15)(10,18)(11,20)(12,17)(13,23)(14,21)(16,24)(19,22)(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=ghi,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, 8095308);