C₂×#520509 #72340057

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

Aliases

direct product

C₂×#520509

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68629 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)(8,10)(12,15)(16,17);(11,12,14,15)(13,16,18,17);(4,6)(8,10)(19,20,23,21)(22,24,26,25);(3,4)(5,6)(7,8,9,10)(11,12)(13,17)(14,15)(16,18)(19,21,23,20)(22,24,26,25);(1,2);(11,13)(12,16)(14,18)(15,17)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72329230, #72340068, #72340074
maximal normal subgroup of	C₂×#72329230, #72340068, #72340074
maximal quotient of	#72340068, C₂×#72337364

GAP

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

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

C2×#520509 #72340057