#187597

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62759	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2212 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,27,29,23)(20,31,32,42)(21,26)(22,25)(24,33,39,35)(28,44)(30,41,34,38)(36,40)(37,46)(43,47)(45,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23)(20,28)(21,34)(22,29)(24,37)(25,27)(26,41)(30,36)(31,44)(32,45)(33,47)(35,46)(38,40)(39,43)(42,48);(4,13)(8,15)(9,16)(10,12)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,39)(26,42)(27,43)(29,45)(30,47)(31,40)(33,36)(34,46)(38,44)(41,48);(2,7)(6,14)(8,15)(9,16)(19,29)(23,27)(28,45)(30,34)(37,43)(38,41)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,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, 94793);