#6343403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343394
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	#62788	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2870 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27)(20,23)(21,25)(22,26)(24,29)(28,31)(30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27);(2,8)(7,15)(9,10)(11,16)(17,20,23,18)(19,28,30,24)(21,25,22,26)(27,29,32,31);(17,21,23,22)(18,25,20,26)(19,29,30,31)(24,32,28,27);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,26,20,25)(19,29,30,31)(24,27,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250599);