#3549

order	128
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 16 points:

(2,4)(6,7)(10,13)(11,12);(1,2,5,4)(3,7,8,6)(9,10,14,13)(11,16,12,15);(9,11,14,12)(10,15,13,16);(1,3,5,8)(2,6,4,7)(9,12,14,11)(10,16,13,15);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,12,15)

Polycyclic presentation:

2^7|b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72272694, #72340807
Fitting subgroup of	#3549⋊C₃, #11714274, #11714278, …
maximal subgroup of	#8491, #8530, #12769, …
maximal normal subgroup of	#8491, #8530, #12769, …
maximal quotient of	#34368, #34415, #34485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2281);