#110228

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
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	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,e^2=hi,f^2=h,g^2=h,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=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17424);