#8115513

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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)(4,5)(6,8)(10,12)(13,21)(15,18)(19,23)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(2,5)(6,7)(10,12)(11,17)(16,20)(19,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,18,17,15)(13,16,21,20)(19,24,23,22)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022709);