#204027

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111223);