#8601580

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8508776);