#232556

order	512
nilpotency class	3
isoclinism	isoclinic class of #232546
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	C₂³×C₄	C₂⁴
prime radical	#37618	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139752);