#1868

order	128
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₂×#7622880 (order 1024) · Inn(G) = #120 · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#120
commutator	C₂³	C₂²×C₄
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂	#385
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#12962, #12963, #12968, …
maximal normal subgroup of	#12962, #12963, #12968, …
maximal quotient of	#8851, #8852, #8861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 600);