#186367

order	512
nilpotency class	3
isoclinism	isoclinic class of #186176
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93563);