C₂×#9122534 #72331072

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#9122534

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 10036 conjugacy classes of subgroups; only the 29525 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290
maximal normal subgroup of	C₂×#72232290

GAP

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

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

C2×#9122534 #72331072