C₂⁵×#535 #72295517

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴×C₄

Aliases

direct product

C₂⁵×#535

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

21023976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 9 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁵×#3484, C₂³×#4312024
maximal normal subgroup of	C₂⁵×#3484, C₂³×#4312024

GAP

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

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

C25×#535 #72295517