C₂×#517592 #72233033

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#517592

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

256501 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72214271, C₂×#72214449, C₂×#72214239, …
maximal normal subgroup of	C₂×#72214271, C₂×#72214449, C₂×#72214239, …
maximal quotient of	D₃×#517592, C₂×D₄×#1438, C₂×#72298107, …

GAP

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

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

C2×#517592 #72233033