C₂×#636800 #72334451

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

Aliases

direct product

C₂×#636800

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

74147 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237099, C₂×#72237104, C₂×#72236761, …
maximal normal subgroup of	C₂×#72237099, C₂×#72237104, C₂×#72236761, …
maximal quotient of	C₂×#72334303, C₂×#72323618

GAP

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

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

C2×#636800 #72334451