C₂×#6342764 #72235068

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#6342764

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

48297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72235062, #72261606, #72261873
maximal normal subgroup of	C₂×#72235062, #72261606, #72261873

GAP

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

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

C2×#6342764 #72235068