#120368

order	512
nilpotency class	3
isoclinism	isoclinic class of #120365
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

386 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fhi,e^c=eh,f^c=fhi,g^c=gi

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, 27564);