#186392

order	512
nilpotency class	3
isoclinism	isoclinic class of #186128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3497	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6,5,8,3,7)(9,10,14,20,15,18,13,19)(11,21,23,34,24,16,29,31)(12,25,27,17,28,33,26,32)(22,36,37,30,38,40,39,35);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,21)(19,31)(20,34)(25,40)(26,37)(27,39)(28,38)(32,35)(33,36);(9,12,15,28)(10,17,18,32)(11,22,24,38)(13,27,14,26)(16,30,21,35)(19,33,20,25)(23,39,29,37)(31,36,34,40);(10,18)(16,21)(17,32)(19,20)(25,33)(30,35)(31,34)(36,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316736, #72316964, #72316966
maximal normal subgroup of	#72316736, #72316964, #72316966

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93588);