#8750093

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657289);