#335881

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28149	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 243077);