C₂×#4034425 #72294577

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

Aliases

direct product

C₂×#4034425

Subgroups

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

28937 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)(9,11)(10,12)(13,14)(15,16,18,20)(17,21,22,19)(23,24)(25,28)(26,29)(27,30);(7,9,10,13)(8,11,12,14)(15,17,18,22)(16,19,20,21)(23,25)(24,27)(26,30)(28,29);(4,6)(8,12)(9,13)(16,20)(19,21)(23,24,26,29)(25,27,30,28);(7,9)(8,11)(10,13)(12,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,29)(27,30);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,17)(16,21)(18,22)(19,20)(23,25)(24,28)(26,30)(27,29);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293940
maximal normal subgroup of	C₂×#72293940
maximal quotient of	C₂×#72293957

GAP

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

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

C2×#4034425 #72294577