C₂×#2081385 #72325284

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

Aliases

direct product

C₂×#2081385

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

42147 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)(13,16)(14,15)(17,18)(19,20)(21,25)(22,23)(24,26);(3,4)(5,6)(7,8)(9,10)(12,14)(13,18)(21,23)(22,25);(4,6)(11,13,15,18)(12,16,14,17)(19,21,24,23)(20,25,26,22);(8,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,26)(24,25);(1,2);(3,4)(5,6)(7,8)(9,10)(11,14,15,12)(13,17,18,16)(19,23,24,21)(20,25,26,22)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232188, C₂×#72232118, C₂×#72232103
maximal normal subgroup of	C₂×#72232188, C₂×#72232118, C₂×#72232103

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

C2×#2081385 #72325284