#343131

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
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	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 631 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:

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

Polycyclic presentation:

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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250327);