#177715

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

632 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28)(35,41)(36,43)(38,46)(39,44)(42,45)(47,48);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(33,34,37,40)(35,38,42,47)(36,44,43,39)(41,48,45,46);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31)(33,35)(34,38)(36,45)(37,42)(39,48)(40,47)(41,43)(44,46);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=ghi,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, 84911);