#191265

order	512
nilpotency class	3
isoclinism	isoclinic class of #191234
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

493 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98461);