#506562

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,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, 413758);