#402171

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1842 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1842 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 256

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309367);