#110870

order	512
nilpotency class	3
isoclinism	isoclinic class of #110864
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18066);