#121949

order	512
nilpotency class	3
isoclinism	isoclinic class of #121711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

420 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=di,e^c=ei,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, 29145);