C₂×#1194991 #72295016

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

Aliases

direct product

C₂×#1194991

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

36827 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)(15,16,19,20)(17,18,21,22)(23,24,27,26)(25,29,30,28);(9,13)(12,14)(16,20)(17,21)(23,24,27,26)(25,28,30,29);(4,6)(8,11)(12,14)(17,21)(18,22)(23,25,27,30)(24,28,26,29);(7,9)(8,12)(10,13)(11,14)(15,17,19,21)(16,18,20,22)(23,26,27,24)(25,28,30,29);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,22)(20,21)(23,25)(24,28)(26,29)(27,30);(1,2)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293918
maximal normal subgroup of	C₂×#72293918
maximal quotient of	C₂×#72293929

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

C2×#1194991 #72295016