#503073

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3339	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 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)(5,6,8,9)(7,11,12,10)(13,14,16,19)(15,23,26,35)(17,30,28,38)(18,31,29,22)(20,34,32,24)(21,37,33,25)(27,43,41,40)(36,42,39,44);(6,9)(10,11)(14,19)(15,24)(17,25)(18,29)(20,35)(21,38)(23,32)(26,34)(27,44)(28,37)(30,33)(36,40)(39,43)(41,42);(1,2,3,4)(13,15,25,42,18,27,28,34,16,26,37,44,29,41,17,24)(14,20,30,40,22,36,33,35,19,32,38,43,31,39,21,23);(5,7,8,12)(6,10,9,11)(14,19)(20,32)(21,33)(22,31)(23,35)(30,38)(36,39)(40,43)

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410269);