C₂×#1737553 #72294593

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

Aliases

direct product

C₂×#1737553

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

31077 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293940
maximal normal subgroup of	C₂×#72293940
maximal quotient of	C₂×#72293967

GAP

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

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

C2×#1737553 #72294593