#395789

order	512
nilpotency class	3
isoclinism	isoclinic class of #395789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14727
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1819 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)(3,10,15,27)(4,9)(5,11)(6,21)(7,12)(8,14,23,16)(13,28,30,24)(17,31,29,22)(18,25)(19,32)(20,26);(1,3)(2,8)(4,17)(5,16)(6,14)(7,15)(9,24)(10,21)(11,27)(12,23)(13,19)(18,30)(20,29)(22,25)(26,28)(31,32);(1,4,5,18,7,20,6,19)(2,9,11,25,12,26,21,32)(3,13,14,29,15,30,16,17)(8,22,10,24,23,31,27,28);(1,3)(2,10)(4,13)(5,14)(6,16)(7,15)(8,21)(9,24)(11,23)(12,27)(17,19)(18,29)(20,30)(22,32)(25,31)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306313, #72306453, #72306465, …
maximal normal subgroup of	#72306313, #72306453, #72306465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302985);