C₂⁵×#534 #72295545

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁵

Aliases

direct product

C₂⁵×#534

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

22712744 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);(16,19)(17,21);(11,12)(13,14)(15,16)(17,20)(18,19)(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×#1082737, C₂³×#1085256
maximal normal subgroup of	C₂⁵×#3484, C₂³×#1082737, C₂³×#1085256
maximal quotient of	C₂²×#72235095, C₂³×#7652795, C₂³×#7642244

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), (16,19)(17,21), (11,12)(13,14)(15,16)(17,20)(18,19)(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×#534 #72295545