#6344178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343485
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	C₄²	C₂⁵
prime radical	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,e^2=i,f^2=i,g^2=hi,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, 6251374);