C₂×#648388 #72325550

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#648388

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

131933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232175, C₂×#72232106, C₂×#72232134
maximal normal subgroup of	C₂×#72232175, C₂×#72232106, C₂×#72232134

GAP

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

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

C2×#648388 #72325550