#8471043

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378239);