#187706

order	512
nilpotency class	3
isoclinism	isoclinic class of #187667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2025 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:

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

Polycyclic presentation:

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

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, 94902);