#190068

order	512
nilpotency class	3
isoclinism	isoclinic class of #190068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

980 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,27,30,23)(20,31,32,46)(21,26)(22,25)(24,33,37,47)(28,41)(29,40,44,36)(34,38)(35,43)(39,48)(42,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,27)(26,36)(31,45)(32,42)(33,43)(34,44)(37,39)(38,40)(41,46)(47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20)(18,24,25,37)(19,28)(21,33)(22,32)(23,35,27,39)(26,31,38,46)(29,43)(30,42)(34,47)(36,45,40,41)(44,48);(3,10)(6,14)(9,16)(13,15)(20,32)(24,37)(28,42)(31,46)(33,47)(35,39)(41,45)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97264);