#187299

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2086 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)(3,10,7,4)(5,8)(6,9)(11,16,14,12)(13,15)(17,18,22,26,23,27,21,35)(19,28,30,24,31,40,36,39)(20,32,33,41,34,25,37,42)(29,44,46,38,47,43,45,48);(1,3,6,7)(2,4,9,10)(5,11,13,14)(8,12,15,16)(17,19,23,31)(18,24,27,39)(20,29,34,47)(21,36,22,30)(25,38,32,48)(26,40,35,28)(33,46,37,45)(41,44,42,43);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,32)(28,43)(30,45)(31,47)(35,42)(36,46)(39,48)(40,44);(1,5,6,13)(2,8,9,15)(3,11,7,14)(4,12,10,16)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94495);