C₂×#1573069 #72294271

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

Aliases

direct product

C₂×#1573069

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

44933 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)(15,16,18,20)(17,19,22,21)(23,24,27,29)(25,28,30,26);(7,8)(9,13)(10,11)(12,14)(15,16,18,20)(17,19,22,21)(23,25,27,30)(24,28,29,26);(4,6)(8,11)(12,13)(15,16,18,20)(17,21,22,19)(23,24)(25,26)(27,29)(28,30);(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,21)(18,20)(19,22)(23,24)(25,28)(26,30)(27,29);(3,4)(5,6)(7,9,10,14)(8,12,11,13)(15,17,18,22)(16,19,20,21)(23,26,27,28)(24,25,29,30);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293931
maximal normal subgroup of	C₂×#72293931
maximal quotient of	C₂×#72279098

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

C2×#1573069 #72294271