C₂×#721684 #72338001

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#721684

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

42529 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)(7,8,9,10)(11,12,14,15)(13,17,18,16)(19,20,22,24)(21,25,26,23);(3,4,5,6)(11,12)(13,16)(14,15)(17,18);(8,10)(12,15)(13,18)(19,20,22,24)(21,25,26,23);(7,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,24)(25,26);(1,2);(3,4)(5,6)(11,13)(12,16)(14,18)(15,17)(19,21,22,26)(20,23,24,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232334, C₂×#72232296, C₂×#72232350
maximal normal subgroup of	C₂×#72232334, C₂×#72232296, C₂×#72232350

GAP

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

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

C2×#721684 #72338001