#187341

order	512
nilpotency class	3
isoclinism	isoclinic class of #187213
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62828	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2106 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)(3,4,10,12)(5,13)(6,8)(7,15)(9,16,14,11)(17,18,22,26,23,27,21,35)(19,28,31,24,30,40,36,39)(20,32,33,41,34,25,37,42)(29,44,47,38,45,43,46,48);(1,3)(2,4)(5,14)(6,10)(7,16)(8,12)(9,15)(11,13)(17,19)(18,24)(20,29)(21,36)(22,31)(23,30)(25,38)(26,40)(27,39)(28,35)(32,48)(33,47)(34,45)(37,46)(41,44)(42,43);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,32)(28,43)(30,45)(31,46)(35,42)(36,47)(39,48)(40,44);(1,5,6,15)(2,7,8,13)(3,9,10,14)(4,11,12,16)(19,30)(24,39)(28,40)(29,45)(31,36)(38,48)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,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, 94537);