#121008

order	512
nilpotency class	3
isoclinism	isoclinic class of #121008
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

378 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=gi,f^c=fi,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, 28204);