#120901

order	512
nilpotency class	3
isoclinism	isoclinic class of #120900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,e^c=eh,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28097);