C₂×#546174 #72324933

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#546174

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49201 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)(13,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26);(3,4,5,6)(19,20,22,23)(21,24,26,25);(4,6)(12,13)(16,17)(20,23)(24,25);(4,6)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,21,22,26)(20,24,23,25);(1,2);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,20)(21,24)(22,23)(25,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232077, C₂×#72232121, C₂×#72232154
maximal normal subgroup of	C₂×#72232077, C₂×#72232121, C₂×#72232154

GAP

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

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

C2×#546174 #72324933