C₂×#649194 #72324127

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#649194

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

44253 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297413, C₂×#72297691, C₂×#72323274
maximal normal subgroup of	C₂×#72297413, C₂×#72297691, C₂×#72323274

GAP

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

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

C2×#649194 #72324127