#273306

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62166	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

803 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180502);