#202131

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	#60962	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,f^2=h,g^2=i,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, 109327);