#251915

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
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	C₂⁵	C₂⁴
prime radical	#2403	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1029 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(22,25)(26,28);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,24)(19,25)(20,23,29,32)(22,27)(26,31)(28,30);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159111);