C₂×#4030235 #72296183

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#4030235

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45871 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231349, C₂×#72232907, C₂×#72277975, …
maximal normal subgroup of	C₂×#72231349, C₂×#72232907, C₂×#72277975, …
maximal quotient of	C₂×#72278089, C₂²×#4030235

GAP

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

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

C2×#4030235 #72296183