#506476

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

824 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)(8,10)(9,12)(13,14,19,32,30,39,29,38,18,23,17,22,16,21,34,44)(15,24,41,36,25,42,31,20,28,43,26,35,27,40,33,37);(1,2,3,4)(5,7,8,12)(6,9,10,11);(2,4)(13,15)(14,20)(16,25)(17,31)(18,28)(19,33)(21,35)(22,40)(23,37)(24,38)(26,29)(27,30)(32,42)(34,41)(36,39)(43,44);(6,10)(9,11)(13,16,18,30)(14,21,23,39)(15,25,28,27)(17,29,19,34)(20,35,37,36)(22,38,32,44)(24,42,43,40)(26,33,41,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413672);