#8926572

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	#55702	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,e^2=h,f^2=g,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, 8833768);