C₂×#845333 #72299331

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

Aliases

direct product

C₂×#845333

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

38977 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)(20,21)(24,25);(4,6)(7,8)(9,10)(11,12,14,15)(13,16,18,17)(19,20,23,21)(22,25,26,24);(4,6)(12,15)(16,17)(19,20,23,21)(22,24,26,25);(4,6)(11,13,14,18)(12,16,15,17)(19,21,23,20)(22,25,26,24);(3,4)(5,6)(11,12)(13,17)(14,15)(16,18)(19,22)(20,24)(21,25)(23,26);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297590, C₂×#72297591, C₂×#72297592
maximal normal subgroup of	C₂×#72297590, C₂×#72297591, C₂×#72297592

GAP

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

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

C2×#845333 #72299331