#483967

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
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

421 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)(4,11,12,15,13,7,14,16)(17,18,21,26,22,28,39,27)(19,29,32,24,30,44,38,43)(20,33,36,25,37,45,47,46)(23,34,41,31,40,48,42,35);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(17,19)(18,23)(20,34)(21,38)(22,30)(24,25)(26,42)(27,41)(28,40)(29,45)(31,47)(32,39)(33,44)(35,36)(37,48)(43,46);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(17,19)(18,24,28,43)(20,35)(21,32)(22,30)(23,25,40,46)(26,44,27,29)(31,37)(33,41,45,42)(34,36)(38,39)(47,48);(2,3)(7,11)(8,9)(15,16)(19,30)(23,40)(24,43)(29,44)(31,35)(32,38)(34,48)(41,42)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,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	#72216730, #72216788, #72216809, …
maximal normal subgroup of	#72216730, #72216788, #72216809, …
maximal quotient of	C₂×#483967, #72318459, #72318489, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391163);