#168831

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76027);