C₂³×#1946 #72307606

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂³×#1946

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
socle	C₂⁵	C₄×D₄
prime radical	C₂⁶	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

22789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#12956, C₂³×#21546
maximal normal subgroup of	C₂³×#12956, C₂³×#21546

GAP

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

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

C23×#1946 #72307606