C₂×#612908 #72334686

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

Aliases

direct product

C₂×#612908

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

66501 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334344, C₂×#72331394, C₂×#72339318
maximal normal subgroup of	C₂×#72334344, C₂×#72331394, C₂×#72339318
maximal quotient of	C₂×#72334386, C₂×#72335140

GAP

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

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

C2×#612908 #72334686