C₂³×#58997 #72289518

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂³×#58997

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3084808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7684106, C₂³×D₄³, C₂³×#8698894, …
maximal normal subgroup of	C₂³×#7684106, C₂³×D₄³, C₂³×#8698894, …

GAP

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

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

C23×#58997 #72289518