#195526

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1135 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214180, #72215973, #72216030
maximal normal subgroup of	#72214180, #72215973, #72216030
maximal quotient of	#72214111, #72215258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102722);