#94849

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁶	C₂³
commutator	C₂³	C₄³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₁

9462 conjugacy classes of subgroups; only the 3218 normal subgroups are shown.

normal subgroups only

This group has 9462 conjugacy classes of subgroups — too many to draw. Only its 3218 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,b^a=bd,c^a=ce,c^b=cf

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2045);