#185522

order	512
nilpotency class	3
isoclinism	isoclinic class of #185517
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	C₂×#3554	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2250 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)(3,8,4,7)(5,9)(6,10)(11,15,12,14)(13,16)(18,22,24,27)(23,29,30,32);(1,3,6,4)(2,7,10,8)(5,12,13,11)(9,15,16,14)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(1,4,6,3)(2,8,10,7)(5,12,13,11)(9,15,16,14)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,d^c=di,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, 92718);