#197251

order	512
nilpotency class	3
isoclinism	isoclinic class of #197002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60558	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104447);