#120807

order	512
nilpotency class	3
isoclinism	isoclinic class of #120804
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

393 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28003);