#497127

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efh,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=ce

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404323);