C₂³×#2315 #72295989

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂³×#2315

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

265045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#34981, C₂³×#34198, C₂³×#35348, …
maximal normal subgroup of	C₂³×#34981, C₂³×#34198, C₂³×#35348, …
maximal quotient of	C₂²×#517417, C₂²×#516395, C₂³×#13877

GAP

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

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

C23×#2315 #72295989