#200132

order	512
nilpotency class	3
isoclinism	isoclinic class of #200128
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	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1072 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)(3,9)(4,12,13,7)(5,11)(6,8)(10,14,16,15)(17,18)(19,24,31,39)(20,32,34,40)(21,26)(22,28)(23,27)(25,35,41,33)(29,43)(30,38,36,42)(37,46)(44,48)(45,47);(1,3,9,2,6,11,5,8)(4,10,14,7,13,16,15,12)(17,19,23,31)(18,24)(20,29,34,45)(21,30,22,36)(25,37)(26,38)(27,39)(28,42)(32,47)(33,44,35,46)(40,43)(41,48);(1,4)(2,7)(3,10)(5,15)(6,13)(8,12)(9,14)(11,16)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,32)(27,41)(28,40)(30,46)(31,45)(36,44)(38,47)(39,48)(42,43);(4,13)(7,12)(10,16)(14,15)(17,21,23,22)(18,26,27,28)(19,30,31,36)(20,33,34,35)(24,38,39,42)(25,40,41,32)(29,44,45,46)(37,43,48,47)

Polycyclic presentation:

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

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, 107328);