#8421856

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

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

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, 8329052);