#188010

order	512
nilpotency class	3
isoclinism	isoclinic class of #187928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62814	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2262 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,27,23,28,21,26)(19,24,32,39,31,38,30,37)(20,25,35,42,34,41,33,40)(29,43,46,47,45,36,44,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,36)(26,37)(27,39)(28,38)(33,44)(34,45)(35,46)(40,47)(41,43)(42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,36)(26,40)(27,42)(28,41)(30,44)(31,45)(32,46)(37,47)(38,43)(39,48);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,32)(20,33,34,35)(24,37,38,39)(25,40,41,42)(29,44,45,46)(36,47,43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^b=fi,g^b=gh,e^c=ei

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, 95206);