#8505993

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	#37343	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 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)(4,6)(5,8)(9,10)(11,18)(12,20)(13,15)(14,16)(17,21)(19,23)(22,24)(25,26,30,34)(27,33,37,31)(28,38,29,32)(35,39,36,40);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(25,27,30,37)(26,31,34,33)(28,35,29,36)(32,39,38,40);(1,2)(3,5)(4,6)(7,8)(25,28,30,29)(26,32,34,38)(27,35,37,36)(31,39,33,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,19)(13,20)(14,22)(15,21)(16,23)(18,24)(25,26)(27,33)(28,32)(29,38)(30,34)(31,37)(35,40)(36,39);(9,13,15,10)(11,16,14,18)(12,20,21,17)(19,23,22,24)(25,27)(26,33)(28,35)(29,36)(30,37)(31,34)(32,40)(38,39);(9,14,15,11)(10,16,13,18)(12,19,21,22)(17,24,20,23)(25,29,30,28)(26,32,34,38)(27,36,37,35)(31,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413189);