#8096953

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	#52292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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)(9,10,16,20)(11,13,17,12)(14,18,21,23)(15,19,24,22)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(2,5)(3,7)(9,11,16,17)(10,12,20,13)(14,15,21,24)(18,22,23,19)(27,33)(31,34)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,17,20,11)(14,22,21,19)(15,18,24,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,13,16,12)(10,11,20,17)(14,19,21,22)(15,23,24,18);(9,14)(10,18)(11,15)(12,22)(13,19)(16,21)(17,24)(20,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,15)(10,19)(11,21)(12,18)(13,23)(14,17)(16,24)(20,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=ghi,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, 8004149);