C₂×#2738366 #72294718

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#2738366

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

32389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(7,8,11,14)(9,12,13,10)(15,16,19,21)(17,18,20,22)(23,24,27,30)(25,26,29,28);(4,6)(7,9,11,13)(8,12,14,10)(15,16,19,21)(17,18,20,22)(23,25)(24,28)(26,30)(27,29);(3,4)(5,6)(7,10)(8,13)(9,14)(11,12)(15,17)(16,18)(19,20)(21,22)(23,24,27,30)(25,26,29,28);(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26)(27,29)(28,30);(7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22)(23,26)(24,29)(25,30)(27,28);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293944
maximal normal subgroup of	C₂×#72293944
maximal quotient of	C₂×#72293984

GAP

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

gap> G := Group((7,8,11,14)(9,12,13,10)(15,16,19,21)(17,18,20,22)(23,24,27,30)(25,26,29,28), (4,6)(7,9,11,13)(8,12,14,10)(15,16,19,21)(17,18,20,22)(23,25)(24,28)(26,30)(27,29), (3,4)(5,6)(7,10)(8,13)(9,14)(11,12)(15,17)(16,18)(19,20)(21,22)(23,24,27,30)(25,26,29,28), (7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,26)(27,29)(28,30), (7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22)(23,26)(24,29)(25,30)(27,28), (1,2));

C2×#2738366 #72294718