#201432

order	512
nilpotency class	3
isoclinism	isoclinic class of #201424
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263908, #72263917, #72263924
maximal normal subgroup of	#72263908, #72263917, #72263924
maximal quotient of	#72264286

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108628);