#6343425

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343418
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	#62800	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3081 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,23)(20,22)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10);(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

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

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, 6250621);