#248920

order	512
nilpotency class	3
isoclinism	isoclinic class of #248899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2408	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2890 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2890 conjugacy classes of subgroups — too many to draw. Only its 188 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 — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156116);