#495374

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

902 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)(15,25)(16,22)(17,32)(18,34)(19,24)(20,28)(21,30)(23,33)(26,40)(27,39)(29,35)(31,38)(36,43)(37,42)(41,44);(2,4)(6,9)(10,11)(13,15,16,26,18,29,30,43,19,27,31,42,33,44,17,28)(14,20,32,41,23,37,38,39,24,36,21,35,34,40,22,25);(5,7,8,12)(6,10,9,11)(13,16,18,30,19,31,33,17)(14,21,23,22,24,32,34,38)(15,26,29,43,27,42,44,28)(20,35,37,25,36,41,40,39);(15,27)(20,36)(25,39)(26,42)(28,43)(29,44)(35,41)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402570);