C₂×#513900 #72331945

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#513900

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁴×C₄²
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43821 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323312, C₂×#72323318, C₂×#72323336
maximal normal subgroup of	C₂×#72323312, C₂×#72323318, C₂×#72323336

GAP

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

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

C2×#513900 #72331945