#345782

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=e,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252978);