C₂×#1574561 #72294983

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1574561

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

46251 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293917, C₂×#72232907
maximal normal subgroup of	C₂×#72293917, C₂×#72232907
maximal quotient of	C₂×#72293951, C₂×#72278217, C₂×#72278224, …

GAP

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

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

C2×#1574561 #72294983