#110967

order	512
nilpotency class	3
isoclinism	isoclinic class of #110954
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

597 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18163);