#111190

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18386);