C₂⁵×D₁₆ #72287461

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁵×D₁₆

Subgroups

	subgroup	G/H
center, socle	C₂⁶	D₈
commutator, Frattini, cosocle	C₈	C₂⁷
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

108373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 26 points:

(1,2);(3,4);(1,2)(3,4)(5,6)(7,8)(9,10)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23);(5,6);(11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16);(7,8);(9,10)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#2177, #72287522, #72287531
maximal normal subgroup of	C₂⁴×#2177, #72287522, #72287531

GAP

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

gap> G := Group((1,2), (3,4), (1,2)(3,4)(5,6)(7,8)(9,10)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23), (5,6), (11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16), (7,8), (9,10));

C25×D16 #72287461