#8460665

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	#53447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=gi,f^2=hi,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, 8367861);