#184978

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#537	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2217 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,e^b=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92174);