C₂×#127645 #72317282

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂×#127645

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#340
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁶	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280528, C₂×#72280248, C₂×#72279574
maximal normal subgroup of	C₂×#72280528, C₂×#72280248, C₂×#72279574

GAP

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

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

C2×#127645 #72317282