C₂×#3076977 #72295025

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

Aliases

direct product

C₂×#3076977

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

37463 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:

(7,8,11,13)(9,12,10,14)(15,16,19,18)(17,21,22,20)(23,24)(25,28)(26,29)(27,30);(7,9)(8,12)(10,11)(13,14)(15,17,19,22)(16,20,18,21)(23,25)(24,27)(26,30)(28,29);(4,6)(8,13)(9,10)(17,22)(20,21)(23,25,26,30)(24,28,29,27);(7,10,11,9)(8,14,13,12)(15,18,19,16)(17,20,22,21)(23,25)(24,28)(26,30)(27,29);(3,4)(5,6)(7,8)(9,14)(10,12)(11,13)(15,17)(16,20)(18,21)(19,22)(23,24)(25,28)(26,29)(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×#72293943

GAP

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

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

C2×#3076977 #72295025