C₂×#72297796 #72298950

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵

Aliases

direct product

C₂×#72297796

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

At least 26368 conjugacy classes of subgroups; only the 418704 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72298104

GAP

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

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

C2×#72297796 #72298950