#97304

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

507 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(5,8)(9,10)(11,19,22,15)(12,23)(13,26)(14,17)(16,24)(18,28)(20,35,37,40)(21,30,39,34)(25,32)(27,31)(29,36,33,38)(41,42)(43,51,54,63)(44,55,57,48)(45,49)(46,50)(47,53,61,58)(52,66)(56,70,72,62)(59,64)(60,67)(65,68)(69,71);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,40)(24,36)(25,37)(26,34)(27,38)(28,39)(31,33)(32,35)(41,43,46,54)(42,47,50,61)(44,52,57,68)(45,58,59,53)(48,60,55,69)(49,63,64,51)(56,71,72,67)(62,66,70,65);(1,3)(2,5)(4,7)(6,8)(9,12,14,25)(10,16,17,31)(11,20,22,37)(13,27,28,24)(15,29,19,33)(18,32,26,23)(21,38,39,36)(30,35,34,40)(41,44)(42,48)(43,52)(45,56)(46,57)(47,60)(49,62)(50,55)(51,65)(53,67)(54,68)(58,71)(59,72)(61,69)(63,66)(64,70)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4500);