#202173

order	512
nilpotency class	3
isoclinism	isoclinic class of #201968
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3192	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109369);