#94789

order	512
nilpotency class	6
isoclinism	isoclinic class of #94787
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7493
commutator	C₄⋊Q₈	C₄²
Frattini, cosocle	#3152	C₂²
prime radical	#532	D₄
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3152 ◁ G
LECS	G ▷ #3152 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

184 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfh,f^b=fh,g^b=ghi,d^c=dgh,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1985);