C₂×#4606592 #72294528

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄³

Aliases

direct product

C₂×#4606592

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

30443 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)(9,12)(10,11)(13,14)(15,16,19,21)(17,18,22,20)(23,24,26,28)(25,27,30,29);(9,13)(12,14)(15,17)(16,20)(18,21)(19,22)(23,24,26,28)(25,29,30,27);(4,6)(8,11)(9,13)(16,21)(17,22)(23,24,26,28)(25,27,30,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,28)(27,30);(7,9)(8,12)(10,13)(11,14)(15,18)(16,22)(17,21)(19,20)(23,25)(24,27)(26,30)(28,29);(1,2)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937
maximal normal subgroup of	C₂×#72293937
maximal quotient of	C₂×#72293975

GAP

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

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

C2×#4606592 #72294528