C₂×#1337681 #72295005

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

Aliases

direct product

C₂×#1337681

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35763 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293917
maximal normal subgroup of	C₂×#72293917
maximal quotient of	C₂×#72293987

GAP

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

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

C2×#1337681 #72295005