#251755

order	512
nilpotency class	3
isoclinism	isoclinic class of #251731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 890 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:

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

Polycyclic presentation:

2^9|b^2=ehi,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,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, 158951);