#187886

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257715, #72286468, #72286487, …
maximal normal subgroup of	#72257715, #72286468, #72286487, …
maximal quotient of	#72257832

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95082);