C₂×#5608921 #72296706

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

Aliases

direct product

C₂×#5608921

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

37621 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)(12,14)(15,16,18,19)(17,20,22,21)(23,24)(25,29)(26,28)(27,30);(4,6)(16,19)(20,21)(23,25)(24,27)(26,30)(28,29);(3,4)(5,6)(9,13)(12,14)(24,28)(27,29);(3,4)(5,6)(7,8)(9,14)(10,11)(12,13)(15,17)(16,20)(18,22)(19,21)(23,25)(24,27)(26,30)(28,29);(3,4)(5,6)(7,9)(8,12)(10,13)(11,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,27)(26,28)(29,30);(1,2)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72230859, C₂×#72278649
maximal normal subgroup of	C₂×#72230859, C₂×#72278649
maximal quotient of	C₂×#72278761, C₂×#72278732, C₂×#72278805

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

C2×#5608921 #72296706