#8425617

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	#34889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4337 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332813);