#355824

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13791
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1181 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)(3,12,15,8)(4,16)(5,7)(6,11)(9,17)(10,20)(13,21,19,25)(14,28,29,32)(18,22)(23,27,26,30)(24,31);(2,7,11,10)(3,13,15,19)(8,12)(9,22,24,16)(14,27,29,30)(21,25)(23,26)(28,32);(1,3)(2,8)(4,14)(5,19)(6,15)(7,21)(9,23)(10,25)(11,12)(13,20)(16,28)(17,30)(18,29)(22,32)(24,26)(27,31);(1,4,5,17,6,18,20,31)(2,9,10,16,11,24,7,22)(3,14,19,30,15,29,13,27)(8,23,25,28,12,26,21,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263020);