#326958

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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	C₂×C₄²	C₂⁴
prime radical	#37608	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234154);