#113142

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60949	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

827 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20338);