#8926532

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	#36721	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833728);