#8508912

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416108);