#288696

order	512
nilpotency class	3
isoclinism	isoclinic class of #288683
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#550
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

536 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:

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

Polycyclic presentation:

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

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, 195892);