#503097

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

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

1700 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

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

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,3)(10,14)(11,15)(12,18)(16,24)(17,22)(19,23)(20,21);(1,2,4,3)(5,6)(7,8);(6,8)(9,10,15,21,12,16,19,22,13,17,23,24,18,20,11,14);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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

maximal subgroup of	#72309928, D₄×#2184, #72310193
maximal normal subgroup of	#72309928, D₄×#2184, #72310193

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410293);