C₂×#521594 #72340058

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#521594

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69189 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72329230, C₂×#72337327, #72340071, …
maximal normal subgroup of	C₂×#72329230, C₂×#72337327, #72340071, …
maximal quotient of	#72340071, C₂×#72337366

GAP

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

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

C2×#521594 #72340058