C₂×#1506690 #72294977

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#1506690

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

50357 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293917
maximal normal subgroup of	C₂×#72293917
maximal quotient of	C₂×#72293947

GAP

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

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

C2×#1506690 #72294977