C₂×#156742 #72295239

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#156742

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

64725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293936, C₂×#72293951, C₂×#72293958, …
maximal normal subgroup of	C₂×#72293936, C₂×#72293951, C₂×#72293958, …

GAP

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

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

C2×#156742 #72295239