C₂×#72250280 #72334767

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72250280

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁴×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6526 conjugacy classes of subgroups; only the 18968 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334421, C₂×#72337376
maximal normal subgroup of	C₂×#72334421, C₂×#72337376

GAP

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

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

C2×#72250280 #72334767