#110992

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

589 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)(3,10,13,14)(4,6,11,8)(7,15,16,12)(17,18,21,27,23,26,22,35)(19,24,30,38,29,37,34,39)(20,25,32,40,31,41,33,46)(28,42,44,48,43,36,45,47);(2,6,9,8)(7,14,16,10)(17,19,22,30,23,29,21,34)(18,24,27,39,26,37,35,38)(20,28,33,44,31,43,32,45)(25,36,40,48,41,42,46,47);(1,3,11,15,5,13,4,12)(2,7,6,14,9,16,8,10)(17,20,21,32,23,31,22,33)(18,25,27,40,26,41,35,46)(19,28,34,45,29,43,30,44)(24,36,39,48,37,42,38,47)

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,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, 18188);