#343105

order	512
nilpotency class	3
isoclinism	isoclinic class of #343099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
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

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

normal subgroups only

This group has 651 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250301);