#3556

order	128
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
prime radical	C₂³	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^7|a^2=f,b^2=fg,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	#72272611, #72272613, #72272617
Fitting subgroup of	#3556⋊C₃
maximal subgroup of	#8495, #8533, #18698, …
maximal normal subgroup of	#8495, #8533, #18698, …
maximal quotient of	#34374, #34421, #34488, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2288);