C₂×#569196 #72294751

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#569196

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

32645 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)(17,20,22,18)(23,24)(25,26)(27,29)(28,30);(9,13)(12,14)(15,17,19,22)(16,20,21,18)(23,25,27,30)(24,28,29,26);(3,4,5,6)(8,11)(9,13)(17,22)(18,20)(24,29)(26,28);(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28)(29,30);(7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,25)(24,26)(27,30)(28,29);(1,2)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232681
maximal normal subgroup of	C₂×#72232681
maximal quotient of	C₂×#72293980

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

C2×#569196 #72294751