#8096938

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	#45604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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)(6,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,22)(23,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(2,5)(3,7)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(27,33)(31,34)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004134);