#237725

order	512
nilpotency class	3
isoclinism	isoclinic class of #237707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60917	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1277 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)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,18)(19,23)(20,30)(21,24)(22,25)(26,28)(27,32)(29,31);(4,5)(7,15)(8,13)(11,14)(17,19,22,29)(18,23,25,31)(20,28,21,27)(24,32,30,26);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,16,15,10)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=eh,f^c=fi

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, 144921);