C₂³×#1386 #72307479

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂³×#1386

Subgroups

	subgroup	G/H
center, socle	C₂⁵	#102
commutator	C₈	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁶	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

14469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(12,16,17,25)(13,19)(14,22)(18,20,23,26)(21,24);(3,4);(7,8,9,10)(11,12)(13,20)(14,23)(15,17)(16,24)(18,22)(19,25)(21,26);(5,6)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#12956, C₂³×#12647, C₂³×#13001
maximal normal subgroup of	C₂³×#12956, C₂³×#12647, C₂³×#13001

GAP

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

gap> G := Group((1,2), (12,16,17,25)(13,19)(14,22)(18,20,23,26)(21,24), (3,4), (7,8,9,10)(11,12)(13,20)(14,23)(15,17)(16,24)(18,22)(19,25)(21,26), (5,6));

C23×#1386 #72307479