#285611

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 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:

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

Polycyclic presentation:

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

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, 192807);