#174464

order	512
nilpotency class	3
isoclinism	isoclinic class of #174093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81660);