C₂×#4423181 #72338005

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#4423181

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

36797 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232299, C₂×#72232314, C₂×#72232350
maximal normal subgroup of	C₂×#72232299, C₂×#72232314, C₂×#72232350

GAP

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

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

C2×#4423181 #72338005