#177704

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

664 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 84900);