C₂×#7625715 #72331672

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7625715

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

124013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72327577
maximal normal subgroup of	C₂×#72327577
maximal quotient of	C₂×#72327626

GAP

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

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

C2×#7625715 #72331672