#230424

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,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, 137620);