#121019

order	512
nilpotency class	3
isoclinism	isoclinic class of #120445
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

401 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,e^c=ei,f^c=fi,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, 28215);