#214751

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35840	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1005 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)(3,10)(4,8)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18)(19,22,26,32)(20,25)(21,23)(24,28)(27,30,29,31);(3,11)(7,15)(10,16)(12,14)(17,19)(18,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,28);(1,3,6,11)(2,7,9,15)(4,12,5,14)(8,16,13,10)(18,23)(21,28)(22,30)(24,25)(27,29)(31,32);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(17,19)(18,22)(20,26)(21,27)(23,30)(24,31)(25,32)(28,29)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cei,f^b=fh,g^b=gh,f^c=fi,g^d=gh

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, 121947);