#213323

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,22,26,23,27,21,37)(19,28,32,38,29,43,31,24)(20,25,35,41,36,42,34,48)(30,44,46,40,33,47,45,39);(2,6)(7,14)(8,13)(9,15)(17,19,21,31,23,29,22,32)(18,24,26,28,27,38,37,43)(20,33,34,46,36,30,35,45)(25,40,41,47,42,39,48,44);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(19,29)(24,38)(28,43)(30,33)(31,32)(39,40)(44,47)(45,46);(3,10)(7,14)(9,15)(11,16)(17,20,22,35,23,36,21,34)(18,25,26,41,27,42,37,48)(19,30,32,46,29,33,31,45)(24,39,28,44,38,40,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=f,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di

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, 120519);