#189291

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

879 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)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,34,36,41)(21,26)(22,27)(23,28)(24,31)(25,35,42,37)(30,38,45,48)(32,40)(33,39)(43,44,47,46);(17,19,21,31,23,33,22,32)(18,24,27,29,28,40,26,39)(20,30,35,44,36,45,37,46)(25,38,41,43,42,48,34,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,41)(28,42)(29,43)(31,44)(32,46)(33,45)(39,47)(40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,34,42,41)(30,44,45,46)(38,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,f^c=fhi,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, 96487);