#121951

order	512
nilpotency class	3
isoclinism	isoclinic class of #121715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
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

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,g^c=ghi

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, 29147);