#189003

order	512
nilpotency class	3
isoclinism	isoclinic class of #188966
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 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:

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

Polycyclic presentation:

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

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, 96199);