C₂×#2615702 #72294546

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#2615702

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

32299 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)(9,10)(11,12)(13,14)(15,16,19,20)(17,18,21,22)(23,24,27,30)(25,26,29,28);(4,6)(7,8,11,12)(9,10,13,14)(16,20)(18,22)(23,25)(24,28)(26,30)(27,29);(3,4,5,6)(8,12)(9,13)(15,17)(16,18)(19,21)(20,22)(23,25)(24,28)(26,30)(27,29);(7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26)(27,29)(28,30);(7,10)(8,9)(11,14)(12,13)(15,18)(16,21)(17,20)(19,22)(23,26)(24,29)(25,30)(27,28);(1,2)

Conjugacy classes & character table

148 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)(9,10)(11,12)(13,14)(15,16,19,20)(17,18,21,22)(23,24,27,30)(25,26,29,28), (4,6)(7,8,11,12)(9,10,13,14)(16,20)(18,22)(23,25)(24,28)(26,30)(27,29), (3,4,5,6)(8,12)(9,13)(15,17)(16,18)(19,21)(20,22)(23,25)(24,28)(26,30)(27,29), (7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26)(27,29)(28,30), (7,10)(8,9)(11,14)(12,13)(15,18)(16,21)(17,20)(19,22)(23,26)(24,29)(25,30)(27,28), (1,2));

C2×#2615702 #72294546