C₂×#152674 #72279239

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#152674

Subgroups

	subgroup	G/H
center, socle	C₂²	#13082
commutator	C₂²×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂×#23909	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#532 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4187 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:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279281, #72280471, #72306296, …
maximal normal subgroup of	#72279281, #72280471, #72306296, …
maximal quotient of	#72279281

GAP

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

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

C2×#152674 #72279239