#190127

order	512
nilpotency class	3
isoclinism	isoclinic class of #190124
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	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

953 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)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,23)(20,30,32,24)(21,25)(22,26)(27,41,43,35)(28,36)(29,37)(31,44,46,38)(33,39)(34,40)(42,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,22,29)(18,23,26,37)(20,27,32,43)(21,33,34,28)(24,35,30,41)(25,39,40,36)(31,45,46,42)(38,48,44,47);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,42)(29,43)(33,45)(34,46)(36,47)(37,41)(39,48)(40,44);(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,42,43,45)(35,47,41,48)

Polycyclic presentation:

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

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, 97323);