#146985

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,8)(9,10,11,12)(13,14)(15,25)(16,23)(17,21)(18,22)(19,33)(20,26)(24,30)(27,39)(28,40)(29,44)(31,38)(32,37)(34,42)(35,41)(36,43);(2,4)(13,15,18,28,19,29,31,42,17,27,32,43,30,41,16,26)(14,20,23,35,24,36,37,39,21,34,38,44,33,40,22,25);(5,6,7,8)(14,21)(20,34)(22,37)(23,38)(24,33)(25,39)(35,44)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54181);