#185873

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2282 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)(9,14,16,11)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,14)(15,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(1,4)(2,3)(5,14)(6,12)(7,16)(8,10)(9,13)(11,15);(1,5,6,15)(2,7,8,13)(3,9,10,16)(4,11,12,14)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,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, 93069);