C₂×#152104 #72279246

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

Aliases

direct product

C₂×#152104

Subgroups

	subgroup	G/H
center, socle	C₂²	#12696
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	C₂×#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#532 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4795 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:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231244, #72279422, #72279506, …
maximal normal subgroup of	C₂×#72231244, #72279422, #72279506, …

GAP

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

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

C2×#152104 #72279246