C₂×#72249537 #72334355

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂×#72249537

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5572744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

464 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂×#72256589, C₂×#72334301, C₂×D₄×#34980, …

GAP

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

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

C2×#72249537 #72334355