C₂×#618316 #72323608

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#618316

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

82605 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323724, #72323737, #72323764, …
maximal normal subgroup of	#72323724, #72323737, #72323764, …
maximal quotient of	C₂×#72298443, C₂×#72334395, C₂×#72298452, …

GAP

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

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

C2×#618316 #72323608