C₂⁵×#124 #72291293

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶×C₄

Aliases

direct product

C₂⁵×#124

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁶×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

1060245 conjugacy classes of subgroups; only the 469973 normal subgroups are shown.

normal subgroups only

This group has 1060245 conjugacy classes of subgroups — too many to draw. Only its 469973 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#2431, C₂⁵×#546, C₂⁵×#535, …
maximal normal subgroup of	C₂⁴×#2431, C₂⁵×#546, C₂⁵×#535, …
maximal quotient of	C₂⁴×#2435, C₂³×#35903, C₂³×#36289

GAP

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

gap> G := Group((12,14)(15,16,19,18)(17,21,22,20), (15,17,19,22)(16,20,18,21), (11,12,13,14)(15,18,19,16)(17,21,22,20), (3,4), (5,6), (7,8), (9,10), (1,2)(3,4)(5,6)(7,8)(9,10));