#174624

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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

674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81820);