#200885

order	512
nilpotency class	3
isoclinism	isoclinic class of #200737
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60798	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1395 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1395 conjugacy classes of subgroups — too many to draw. Only its 246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317711, #72317775, #72317810, …
maximal normal subgroup of	#72317711, #72317775, #72317810, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108081);