C₂×#152119 #72283175

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#152119

Subgroups

	subgroup	G/H
center, socle	C₂²	#12697
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	#532	C₂⁴
prime radical	C₂×#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,6,12,8,11,19,27)(5,14,16,28,15,9,22,24)(7,10,18,26,21,13,20,29)(17,23,32,33,30,25,31,34);(3,5)(4,9)(6,16)(7,17)(8,15)(10,25)(11,14)(12,28)(13,23)(18,32)(19,22)(20,31)(21,30)(24,27)(26,33)(29,34);(4,10)(5,15)(6,18)(7,21)(9,23)(11,13)(14,25)(16,31)(19,20)(22,32)(24,28)(26,29)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231259, C₂×#72231262, C₂×#72231252, …
maximal normal subgroup of	C₂×#72231259, C₂×#72231262, C₂×#72231252, …

GAP

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

gap> G := Group((1,2), (3,4,6,12,8,11,19,27)(5,14,16,28,15,9,22,24)(7,10,18,26,21,13,20,29)(17,23,32,33,30,25,31,34), (3,5)(4,9)(6,16)(7,17)(8,15)(10,25)(11,14)(12,28)(13,23)(18,32)(19,22)(20,31)(21,30)(24,27)(26,33)(29,34), (4,10)(5,15)(6,18)(7,21)(9,23)(11,13)(14,25)(16,31)(19,20)(22,32)(24,28)(26,29));

C2×#152119 #72283175