#189553

order	512
nilpotency class	3
isoclinism	isoclinic class of #189535
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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:

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

Polycyclic presentation:

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

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, 96749);