#238999

order	512
nilpotency class	3
isoclinism	isoclinic class of #238990
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

822 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:

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

Polycyclic presentation:

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

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, 146195);