C₂×#3333145 #72332674

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

Aliases

direct product

C₂×#3333145

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

41141 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323402, C₂×#72232349
maximal normal subgroup of	C₂×#72323396, C₂×#72323402, C₂×#72232349

GAP

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

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

C2×#3333145 #72332674