#3568

order	128
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	Aut(G) order 6144
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
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

479 conjugacy classes of subgroups; only the 378 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 378 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^7|a^2=f,c^2=f,d^2=fg,e^2=fg,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	#72272685, #72272703
Fitting subgroup of	#3568⋊C₃
maximal subgroup of	#8514, #19038, #19039, …
maximal normal subgroup of	#8514, #19038, #19039, …
maximal quotient of	#34405, #34448, #34451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2300);