C₂×#1515693 #72295031

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

Aliases

direct product

C₂×#1515693

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

44255 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)(17,18)(19,20)(21,22)(23,24,27,26)(25,28,30,29);(8,11)(12,14)(17,21)(18,22)(23,25,27,30)(24,28,26,29);(4,6)(8,11)(9,13)(16,20)(18,22)(23,25,27,30)(24,29,26,28);(7,9,10,13)(8,12,11,14)(15,17)(16,18)(19,21)(20,22)(23,26,27,24)(25,29,30,28);(3,4)(5,6)(7,8)(9,14)(10,11)(12,13)(15,18)(16,17)(19,22)(20,21)(23,25)(24,29)(26,28)(27,30);(1,2)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293918
maximal normal subgroup of	C₂×#72293918
maximal quotient of	C₂×#72293947

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

C2×#1515693 #72295031