#244559

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1076 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,13)(4,15)(5,8)(6,11)(7,12)(9,14)(10,16)(20,27)(22,28)(23,30)(26,31);(2,8)(5,6)(9,13)(10,15)(11,12)(14,16)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3,7,4)(2,9,12,10)(5,14,6,16)(8,15,11,13)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4,7,3)(2,10,12,9)(5,14,6,16)(8,15,11,13)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,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, 151755);