#121647

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

562 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,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, 28843);