#508005

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	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^c=de,f^c=fh,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, 415201);