#168194

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

837 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 837 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75390);