#331929

order	512
nilpotency class	3
isoclinism	isoclinic class of #331927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239125);