#347054

order	512
nilpotency class	3
isoclinism	isoclinic class of #347042
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#62826	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2289 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254250);