C₂³×#2275 #72289386

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄² · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂³×#2275

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂	C₂⁵×C₄²
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

201029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#35921, C₂³×#34290, C₂²×#4928968, …
maximal normal subgroup of	C₂³×#35921, C₂³×#34290, C₂²×#4928968, …
maximal quotient of	C₂³×D₄×#95, C₂²×#513587, C₂²×#513714

GAP

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

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

C23×#2275 #72289386