#190069

order	512
nilpotency class	3
isoclinism	isoclinic class of #190068
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×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

936 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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)(4,7)(5,8)(10,15,16,13)(12,14)(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);(4,12)(7,14)(10,16)(13,15)(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);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(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);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(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,d^2=i,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, 97265);