#265369

order	512
nilpotency class	3
isoclinism	isoclinic class of #265359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62270	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

912 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,2,5,15,6,9,4,8)(3,7,13,16,14,11,12,10)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,2)(3,11)(4,8)(5,15)(6,9)(7,14)(10,13)(12,16)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30);(1,3,5,13,6,14,4,12)(2,7,15,16,9,11,8,10)

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172565);