#8125676

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8032872);