C₂×#2520385 #72278624

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#2520385

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

33327 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72230894, C₂×#72293948
maximal normal subgroup of	C₂×#72230894, C₂×#72293948
maximal quotient of	C₂×#72293957, C₂×#72293959, C₂×#72278160

GAP

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

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

C2×#2520385 #72278624