#200887

order	512
nilpotency class	3
isoclinism	isoclinic class of #200737
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	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,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

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306334, #72306366, #72306879, …
maximal normal subgroup of	#72306334, #72306366, #72306879, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108083);