#6343487

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	C₂×#3494	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 6250683);