#230426

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,9,5,7,12,6)(3,8,10,15,11,14,16,13)(17,18)(19,29,35,44)(20,36)(21,39)(22,26)(23,28)(24,33,42,34)(25,38)(27,30)(31,32,46,48)(37,43)(40,41,47,45);(2,6)(3,10)(4,12)(7,9)(11,16)(13,15)(17,19,30,33,23,35,22,34)(18,24,27,29,28,42,26,44)(20,31,38,45,39,46,37,41)(21,40,43,48,36,47,25,32);(2,7)(3,11)(6,9)(10,16)(17,20)(18,25)(19,31)(21,27)(22,37)(23,39)(24,32)(26,36)(28,43)(29,40)(30,38)(33,45)(34,41)(35,46)(42,48)(44,47);(1,3,4,10,5,11,12,16)(2,8,9,15,7,14,6,13)(17,21,22,25,23,36,30,43)(18,20,26,37,28,39,27,38)(19,32,33,40,35,48,34,47)(24,41,29,31,42,45,44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137622);