#483998

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33746	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286247, #72286482, #72286506, …
maximal normal subgroup of	#72286247, #72286482, #72286506, …
maximal quotient of	#72343443, #72343450, #72343469, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391194);