#327814

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 894 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,d^2=i,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fh

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, 235010);