#506526

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25777	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

815 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)(8,10)(9,12)(13,14,20,25,38,28,21,29,18,23,19,24,17,15,26,30)(16,31,36,43,33,44,39,22,34,42,35,27,37,32,40,41);(5,7,8,12)(6,9,10,11)(13,15,20,30,38,14,21,25,18,28,19,29,17,23,26,24)(16,32,36,41,33,31,39,43,34,44,35,22,37,42,40,27);(2,4)(13,16)(14,22)(15,27)(17,33)(18,34)(19,39)(20,40)(21,35)(23,41)(24,32)(25,44)(26,36)(28,43)(29,31)(30,42)(37,38);(6,10)(9,11)(13,17,18,38)(14,15,23,28)(16,33,34,37)(19,21,20,26)(22,27,41,43)(24,29,25,30)(31,44,42,32)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413722);