#121294

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28490);