#97361

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

504 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(2,5,6,7)(9,10,12,16,13,15,21,23)(11,17,18,24,19,14,20,22)(25,26)(27,36)(28,40,42,49)(29,43)(30,34)(31,35)(32,38)(33,41,50,45)(37,52,55,46)(39,47)(44,48)(51,54,56,53);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,31,39)(26,32,35,48)(28,37,42,55)(29,44,30,38)(33,46,50,52)(34,47,43,36)(40,51,49,56)(41,54,45,53);(3,8)(5,7)(25,28)(26,33)(27,37)(29,41)(30,45)(31,42)(32,46)(34,49)(35,50)(36,51)(38,53)(39,55)(40,43)(44,54)(47,56)(48,52)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=hi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4557);