#185305

order	512
nilpotency class	3
isoclinism	isoclinic class of #185255
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 92501);