#152928

order	512
nilpotency class	5
isoclinism	isoclinic class of #149911
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#127 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

467 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=e,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,f^b=fgh,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60124);