#413829

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321025);