#6343492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343485
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62814	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,f^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250688);