#308311

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 215507);