C₂×#648010 #72331840

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

Aliases

direct product

C₂×#648010

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

39517 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)(12,15)(16,18)(21,22)(24,26);(7,8)(9,10)(19,20)(21,24)(22,26)(23,25);(11,12)(13,16)(14,15)(17,18)(19,21,23,22)(20,24,25,26);(1,2);(4,6)(11,12)(13,16)(14,15)(17,18)(19,22,23,21)(20,24,25,26);(3,4)(5,6)(11,13,14,17)(12,16,15,18)(19,20)(21,24)(22,26)(23,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297936, C₂×#72323324, C₂×#72323330
maximal normal subgroup of	C₂×#72297936, C₂×#72323324, C₂×#72323330

GAP

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

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

C2×#648010 #72331840