C₂³×#34256 #72324537

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂³×#34256

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5979656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7631286
maximal normal subgroup of	C₂³×#7631286

GAP

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

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

C23×#34256 #72324537