C₂×#4928947 #72302603

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#4928947

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68937 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280357, C₂×#72280318, C₂×#72280561
maximal normal subgroup of	C₂×#72280357, C₂×#72280318, C₂×#72280561

GAP

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

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

C2×#4928947 #72302603