#387805

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14495
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

604 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=e,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295001);