C₂×#4308709 #72339961

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#4308709

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68125 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72337224, #72339988, #72339996
maximal normal subgroup of	C₂×#72337224, #72339988, #72339996
maximal quotient of	#72339996, C₂×#72337370

GAP

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

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

C2×#4308709 #72339961