#188022

order	512
nilpotency class	3
isoclinism	isoclinic class of #187944
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

2248 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)(17,18,22,26,23,27,21,35)(19,24,31,37,30,38,29,45)(20,25,34,39,33,40,32,46)(28,41,44,48,43,36,42,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,36)(26,37)(27,38)(32,42)(33,43)(34,44)(35,45)(39,47)(40,41)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,28)(21,32)(22,34)(23,33)(24,36)(26,39)(27,40)(29,42)(30,43)(31,44)(35,46)(37,47)(38,41)(45,48);(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,32,33,34)(24,37,38,45)(25,39,40,46)(28,42,43,44)(36,47,41,48)

Polycyclic presentation:

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

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, 95218);