#395530

order	512
nilpotency class	3
isoclinism	isoclinic class of #395530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14725
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3019	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302726);