#230412

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

559 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137608);