#200292

order	512
nilpotency class	3
isoclinism	isoclinic class of #200146
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 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)(3,5)(4,12,14,8)(6,11)(7,9)(10,13,16,15)(17,18,22,28,23,26,21,27)(19,24,32,39,30,40,31,38)(20,33,36,25,34,42,35,41)(29,43,46,37,44,47,45,48);(1,3,5,2,7,11,6,9)(4,10,13,8,14,16,15,12)(17,19,21,32,23,30,22,31)(18,24,28,38,26,40,27,39)(20,29,36,45,34,44,35,46)(25,37,33,47,41,48,42,43);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,12)(11,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,37)(26,41)(27,42)(28,33)(30,44)(31,46)(32,45)(38,47)(39,43)(40,48);(1,5,7,6)(2,3,9,11)(4,13,14,15)(8,10,12,16)(18,26)(19,30)(25,41)(27,28)(29,44)(31,32)(33,42)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,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, 107488);