#97185

order	512
nilpotency class	3
isoclinism	isoclinic class of #97164
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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

648 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

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, 4381);