C₂×#2112563 #72294545

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

Aliases

direct product

C₂×#2112563

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

32973 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293949

GAP

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

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

C2×#2112563 #72294545