#495523

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402719);