#174612

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

593 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=h,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, 81808);