#506275

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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	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)(13,14,17,22)(15,19,24,33)(16,20,29,36)(18,30,28,23)(21,38,37,27)(25,35,39,31)(26,43,40,34)(32,41,44,42);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,24)(18,26)(20,35)(21,32)(22,33)(23,34)(27,42)(28,40)(29,39)(30,43)(31,36)(37,44)(38,41);(2,4)(14,20)(15,24)(16,27)(18,28)(19,31)(21,23)(22,36)(25,41)(29,38)(30,37)(32,43)(33,35)(34,44)(39,42);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=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, 413471);