#506437

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 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)(5,6,9,12)(7,11,8,10)(13,14,17,23)(15,21,27,20)(16,29,19,22)(18,24,25,34)(26,35,30,37)(28,38,41,36)(31,43,33,39)(32,40,42,44);(5,7,9,8)(6,10,12,11)(13,15,25,41,16,30,33,32,17,27,18,28,19,26,31,42)(14,20,34,38,29,35,39,44,23,21,24,36,22,37,43,40);(2,4)(13,15,17,27)(14,21,23,20)(16,26,19,30)(18,32,25,42)(22,37,29,35)(24,40,34,44)(28,31,41,33)(36,39,38,43);(5,8,9,7)(6,11,12,10)(13,16,17,19)(14,22,23,29)(15,26,27,30)(18,31,25,33)(20,35,21,37)(24,39,34,43)(28,32,41,42)(36,40,38,44)

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 413633);