C₂×#531613 #72298730

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#531613

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

75609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233565, C₂×#72233563, C₂×#72233578, …
maximal normal subgroup of	C₂×#72233565, C₂×#72233563, C₂×#72233578, …
maximal quotient of	C₂×#72298547, C₂²×#531613

GAP

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

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

C2×#531613 #72298730