#3567

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

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

normal subgroups only

This group has 475 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) = 64

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

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

Polycyclic presentation:

2^7|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	#72272595, #72272601, #72272603, …
Fitting subgroup of	#3567⋊C₃
maximal subgroup of	#8515, #19040, #19041, …
maximal normal subgroup of	#8515, #19040, #19041, …
maximal quotient of	#34405, #34449, #34450, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2299);