C₂⁵×#120 #72255579

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁹

Aliases

C₂⁵×#120

	subgroup	G/H
center, socle	C₂⁷	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

5215477 conjugacy classes of subgroups; only the 517845 normal subgroups are shown.

normal subgroups only

This group has 5215477 conjugacy classes of subgroups — too many to draw. Only its 517845 normal subgroups are shown.

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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 18 points:

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

448 conjugacy classes. The character table is too large to display.

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4), (5,6), (7,8), (12,14)(16,18), (9,10), (16,18), (1,2)(3,4)(5,6)(7,8)(9,10), (11,12)(13,14)(15,16)(17,18));