#8325456

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	#59307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8232652);