#200849

order	512
nilpotency class	3
isoclinism	isoclinic class of #200701
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306366, #72306370, #72306372
maximal normal subgroup of	#72306366, #72306370, #72306372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108045);