#251756

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	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 804 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^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,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, 158952);