#450249

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15961
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

2^9|a^2=efh,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,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, 357445);