#120740

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	#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:

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

Polycyclic presentation:

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

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, 27936);