C₂×#2309611 #72294516

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

Aliases

direct product

C₂×#2309611

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

37203 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937, C₂×#72278043
maximal normal subgroup of	C₂×#72293937, C₂×#72278043
maximal quotient of	C₂×#72293961, C₂×#72278117

GAP

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

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

C2×#2309611 #72294516