#196351

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2433	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1213 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103547);