#1833

order	128
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	Aut(G) = #72236881 (order 1024) · Inn(G) = #117 · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#117
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#382
prime radical	C₂³	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^7|a^2=f,b^2=e,c^2=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg,f^c=fg

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#12826, #12827, #12828, …
maximal normal subgroup of	#12826, #12827, #12828, …
maximal quotient of	#8780, #8781, #8782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 565);