C₂×#1418505 #72332709

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

Aliases

direct product

C₂×#1418505

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48425 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)(20,23)(21,25);(4,6)(7,8)(9,10)(11,12,15,16)(13,17,18,14)(19,20,22,23)(21,24,25,26);(4,6)(19,20,22,23)(21,26,25,24);(1,2);(3,4)(5,6)(11,13,15,18)(12,14,16,17)(19,21,22,25)(20,24,23,26);(11,14)(12,13)(15,17)(16,18)(19,20)(21,26)(22,23)(24,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297762, C₂×#72323407, C₂×#72323384
maximal normal subgroup of	C₂×#72297762, C₂×#72323407, C₂×#72323384

GAP

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

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

C2×#1418505 #72332709