C₂×#3069070 #72325484

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

Aliases

direct product

C₂×#3069070

Subgroups

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

43435 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232080, C₂×#72232173
maximal normal subgroup of	C₂×#72232077, C₂×#72232080, C₂×#72232173

GAP

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

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

C2×#3069070 #72325484