#238921

order	512
nilpotency class	3
isoclinism	isoclinic class of #238921
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	#60916	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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)(3,10)(4,12)(5,13)(6,8)(7,11)(9,15)(14,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,28)(29,31)(30,32);(1,3)(2,7)(4,9)(5,14)(6,12)(8,16)(10,13)(11,15)(18,23)(19,28)(20,21)(24,31)(25,26)(29,30);(1,4,5,7,6,10,15,16)(2,3,9,11,8,12,13,14)(17,19,20,29,22,30,21,28)(18,24,26,32,25,31,23,27);(2,8)(4,10)(7,16)(9,13)(18,25)(23,26)(24,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146117);