C₂×#8422218 #72331697

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

Aliases

direct product

C₂×#8422218

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 6576 conjugacy classes of subgroups; only the 29637 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72261136

GAP

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

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

C2×#8422218 #72331697