#307334

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=fhi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 214530);