#97363

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
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	C₄³	C₂³
prime radical	#46236	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 666 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,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, 4559);