#187217

order	512
nilpotency class	3
isoclinism	isoclinic class of #187213
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2234 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)(3,4,10,12)(5,13)(6,8)(7,15)(9,16,14,11)(17,18)(19,27,30,24)(20,25)(21,23)(22,26)(28,31,29,32);(1,3)(2,4)(5,14)(6,10)(7,16)(8,12)(9,15)(11,13)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19,22,30)(18,24,26,27)(20,28,21,29)(23,31,25,32);(1,5,6,15)(2,7,8,13)(3,9,10,14)(4,11,12,16)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,32,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94413);