C₂×#4597888 #72294572

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄³

Aliases

direct product

C₂×#4597888

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

43483 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)(9,12)(10,11)(13,14)(15,16,18,19)(17,20,21,22)(23,24)(25,26)(27,29)(28,30);(9,13)(12,14)(16,19)(17,21)(23,24,27,29)(25,26,30,28);(4,6)(8,11)(9,13)(16,19)(20,22)(23,25,27,30)(24,28,29,26);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,30)(28,29);(7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,26)(24,25)(27,28)(29,30);(1,2)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912, C₂×#72293917
maximal normal subgroup of	C₂×#72293912, C₂×#72293917
maximal quotient of	C₂×#72293972, C₂×#72293976

GAP

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

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

C2×#4597888 #72294572