#1308

order	128
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	Aut(G) = #72231846 (order 3072) · Inn(G) = #95 · Out(G) = C₂×#41⋊C₃
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#95
commutator, prime radical	C₂³	C₄²
Frattini, cosocle	C₂×C₄²	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

61 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^7|a^2=d,b^2=e,d^2=fg,e^2=f,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#1308⋊C₃, C₂×#1308⋊C₃, #8126⋊C₃, …
Fitting subgroup of	#1308⋊C₃
maximal subgroup of	C₂×#1308, #8124, #8125, …
maximal normal subgroup of	C₂×#1308, #8124, #8125, …
maximal quotient of	#7033, #7041, #7048, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 40);