#333170

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

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

normal subgroups only

This group has 621 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:

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

Polycyclic presentation:

2^9|a^2=ei,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, 240366);