#110993

order	512
nilpotency class	3
isoclinism	isoclinic class of #110956
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

569 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:

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

Polycyclic presentation:

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

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, 18189);