#201641

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	C₂×C₄²	C₂⁴
prime radical	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,g^2=hi,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	#72317975, #72318561, #72318627
maximal normal subgroup of	#72317975, #72318561, #72318627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108837);