#495370

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 402566);