C₂×#2130495 #72325097

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

Aliases

direct product

C₂×#2130495

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

49355 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)(11,12,15,13)(14,16,18,17)(19,20,23,22)(21,25,26,24);(3,4)(5,6)(20,22)(21,26);(7,8)(9,10)(12,13)(14,18)(20,22)(24,25);(4,5)(7,8,9,10)(11,13,15,12)(14,16,18,17)(19,21,23,26)(20,24,22,25);(1,2);(3,5,6,4)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22,23,20)(21,25,26,24)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232078, C₂×#72232158, C₂×#72232134, …
maximal normal subgroup of	C₂×#72232078, C₂×#72232158, C₂×#72232134, …

GAP

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

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

C2×#2130495 #72325097