#177708

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
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

640 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)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32)(33,34,38,42)(35,43,45,48)(36,41,37,40)(39,46,47,44);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(18,23)(20,21)(24,31)(25,26)(27,32)(29,30);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19,22,28)(18,24,25,32)(20,30,21,29)(23,31,26,27)(33,35)(34,39)(36,44)(37,46)(38,45)(40,43)(41,48)(42,47);(2,4)(7,10)(8,13)(9,16)(18,25)(19,28)(23,26)(29,30)(33,36,38,37)(34,40,42,41)(35,44,45,46)(39,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84904);