#178417

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46089	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1226 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 85613);