#180456

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

685 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,44,48,45);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,43)(42,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,29)(26,31)(27,32)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45);(2,8)(6,13)(7,14)(12,15)(18,23)(19,26)(24,28)(27,29)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87652);