C₂×#516324 #72333126

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂×#516324

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

253981 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323398, C₂×#72323411, C₂×#72240162, …
maximal normal subgroup of	C₂×#72323398, C₂×#72323411, C₂×#72240162, …
maximal quotient of	C₂×#72316956, C₂×#72334401, C₂×#72335144, …

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

C2×#516324 #72333126