#187314

order	512
nilpotency class	3
isoclinism	isoclinic class of #187177
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	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2086 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)(3,10,7,4)(5,13)(6,9)(8,15)(11,12,14,16)(17,18,22,27,23,28,21,26)(19,29,33,24,32,39,31,38)(20,25,34,40,35,41,36,42)(30,44,45,48,46,43,47,37);(1,3,6,7)(2,4,9,10)(5,14,15,11)(8,16,13,12)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,37)(26,39)(27,29)(28,38)(34,47)(35,46)(36,45)(40,43)(41,48)(42,44);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,13)(15,16)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,40)(28,41)(29,43)(31,47)(32,46)(33,45)(38,48)(39,44);(1,5,6,15)(2,8,9,13)(3,11,7,14)(4,12,10,16)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94510);