#190111

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

964 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)(21,25)(22,26)(24,33)(28,36,43,41)(29,39,34,37)(32,44)(35,40)(38,46)(42,47,45,48);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,23)(20,28)(21,34)(22,30)(24,36)(25,39)(26,27)(29,35)(31,41)(32,45)(33,43)(37,40)(38,48)(42,46)(44,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20)(18,24,26,31)(19,28)(21,32)(22,33)(23,36,27,41)(25,38,40,44)(29,42)(30,43)(34,45)(35,46)(37,47,39,48);(17,21,22,35)(18,25,26,40)(19,29,30,34)(20,32,33,46)(23,37,27,39)(24,38,31,44)(28,42,43,45)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fh,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, 97307);