#8109969

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

4270 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=gi,f^2=g,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, 8017165);