C₂×#612485 #72298778

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#612485

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

72497 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)(12,14)(15,16,19,17)(18,20,22,21);(4,6)(8,10)(11,12)(13,14)(18,22)(20,21);(4,6)(7,8,9,10)(15,17,19,16)(18,20,22,21);(7,8)(9,10);(3,4)(5,6)(7,8)(9,10)(15,18)(16,20)(17,21)(19,22);(1,2)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233548, C₂×#72233549, C₂×#72279803, …
maximal normal subgroup of	C₂×#72233548, C₂×#72233549, C₂×#72279803, …
maximal quotient of	C₂×#72317648, C₂×#72334426, C₂²×#612485

GAP

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

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

C2×#612485 #72298778