C₂⁵×B(4,𝔽₂) #72279273

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁹

Aliases

direct product

C₂⁵×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂⁶	#120
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

7994168 conjugacy classes of subgroups; only the 520296 normal subgroups are shown.

normal subgroups only

This group has 7994168 conjugacy classes of subgroups — too many to draw. Only its 520296 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) = 18
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

automorphism group of

#7122, #7123, #7250, …

GAP

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

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