#339848

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 647 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247044);