#251787

order	512
nilpotency class	3
isoclinism	isoclinic class of #251761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 158983);