#333159

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

597 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240355);