C₂×#554184 #72333851

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#554184

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49841 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327626
maximal normal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327626

GAP

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

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

C2×#554184 #72333851