#333190

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	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 609 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=hi,d^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, 240386);