#446016

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15828
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353212);