#288712

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 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)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(2,8,11,10)(3,13,14,16)(4,12)(9,15)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(1,3)(2,9)(4,10)(5,13)(6,16)(7,14)(8,12)(11,15)(17,19,20,27)(18,23,22,25)(21,29,28,26)(24,31,30,32);(1,4,5,15,7,12,6,9)(2,3,10,13,11,14,8,16)(18,22)(19,27)(24,30)(26,29)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=ei,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, 195908);