C₂×#3071761 #72294532

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

Aliases

direct product

C₂×#3071761

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

37853 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)(9,12,14,13)(15,16,19,21)(17,18,22,20)(23,24,27,26)(25,29,30,28);(7,8,10,11)(9,13,14,12)(15,17)(16,20)(18,21)(19,22)(23,25)(24,28)(26,29)(27,30);(4,6)(8,11)(9,14)(16,21)(17,22)(23,24,27,26)(25,28,30,29);(7,9,10,14)(8,12,11,13)(15,18,19,20)(16,22,21,17)(23,26,27,24)(25,28,30,29);(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,21)(20,22)(23,25)(24,28)(26,29)(27,30);(1,2)

Conjugacy classes & character table

148 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×#72293945, C₂×#72293958

GAP

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

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

C2×#3071761 #72294532