#202175

order	512
nilpotency class	3
isoclinism	isoclinic class of #201978
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	#60965	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 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:

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

Polycyclic presentation:

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

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, 109371);