C₂×#1514961 #72325573

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

Aliases

direct product

C₂×#1514961

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

42161 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232172, C₂×#72232080, C₂×#72232102
maximal normal subgroup of	C₂×#72232172, C₂×#72232080, C₂×#72232102

GAP

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

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

C2×#1514961 #72325573