#8180458

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	#34957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4555 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087654);