C₂×#1145630 #72325528

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1145630

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

43527 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232093, C₂×#72232080, C₂×#72232179
maximal normal subgroup of	C₂×#72232093, C₂×#72232080, C₂×#72232179

GAP

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

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

C2×#1145630 #72325528