#288616

order	512
nilpotency class	3
isoclinism	isoclinic class of #288611
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	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,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,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195812);