#508045

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1134 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=de,f^c=fh,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, 415241);