#8097202

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	#36004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4431 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8004398);